hudm4122 probability and statistical inference february 2, 2015
TRANSCRIPT
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HUDM4122Probability and Statistical Inference
February 2, 2015
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In the last class
• Covariance• Correlation• Scatterplots• Simple linear regression
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Questions? Comments?
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Today
• Ch. 4.1-4.3 in Mendenhall, Beaver, & Beaver
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Today
• Probability in Statistics• Events and the Sample Space• Calculating Probabilities Using Simple Events
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I have a coin in my hand
• What is the probability that I will get heads?
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I have a coin in my hand
• What is the probability that I will get heads?– 50%
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I have a coin in my hand
• What is the probability that I will get tails?
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I have a coin in my hand
• What is the probability that I will get tails?– 50%
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Flipping the Coin
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Now that I’ve flipped the coin
• What is the probability that I will get heads if I flip it again?
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Flipping the coin 10 times
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Flipping the coin 10 times
• What are the odds that I will get heads the next time?
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Flipping the coin 10 times
• What are the odds that I will get heads the next time?
• It depends on whether it’s a fair coin– E.g. a coin where the probability of heads is
exactly 50%
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But if you get heads 10 times in a row
• It’s probably not a fair coin
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Probability of heads 1 time, if it’s a fair coin
• ½, or 50%, or 0.5
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Probability of heads 2 times, if it’s a fair coin
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Probability of heads 2 times, if it’s a fair coin
• 4 cases
• T, H• H, T• T, T• H, H
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Probability of heads 2 times, if it’s a fair coin
• 4 cases
• T, H• H, T• T, T• H, H
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Probability of heads 2 times, if it’s a fair coin
• 4 cases
• T, H• H, T• T, T• H, H
• ¼ , or 25%, or 0.25
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Probability of heads 3 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
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Probability of heads 3 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
• 1/8 , or 12.5%, or 0.125
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Probability of heads 4 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
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Probability of heads 4 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
• 1/16 , or 0.0625
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Probability of heads 5 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
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Probability of heads 5 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
• 1/32, or 0.03125
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Probability of heads 6 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
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Probability of heads 6 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
• 1/64, or 0.015065
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Probability of heads 7 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
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Probability of heads 7 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
• 1/128, or 0.0075325
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Probability of heads 8 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
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Probability of heads 8 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
• 1/256, or about 0.0038
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Probability of heads 9 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
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Probability of heads 9 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
• 1/512, or about 0.002
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Probability of heads 10 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
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Probability of heads 10 times, if it’s a fair coin
• How many cases?• How many of these cases are heads?
• 1/1024, or about 0.001
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Who thinks this is a fair coin?
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Who thinks this is a fair coin?
• It’s really improbable that it is
• About 1 in 1000
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Who thinks this is a fair coin?
• It’s really improbable that it is
• About 1 in 1000
• Of course, weirder things have happened…– My daughter is 1 in 10,000,000,000, so…
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Questions? Comments?
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Quick non-on-the-test digression
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We just did our first statistical test of the semester…
• The sign test
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We just did our first statistical test of the semester…
• The sign test– A special case of it, anyways
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The Sign Test
• Formulas and theory are out of this class’s scope
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The Sign Test
• Formulas and theory are out of this class’s scope
• But the big idea is• You have an event that you think happens 50%
of the time• But does it really occur 50% of the time?
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End quick non-on-the-test digression
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So, that coin
• Every time you flipped that coin it was a simple event– An outcome that is received on a single repetition
of an experiment
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So, that coin
• Every time you flipped that coin it was a simple event– An outcome that is received on a single repetition
of an experiment
• So when I flipped the coin ten times, it represented 10 simple events
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So, that coin
• Every time you flipped that coin it was a simple event– An outcome that is received on a single repetition
of an experiment
• So when I flipped the coin ten times, it represented 10 simple events
• An event is a set of connected simple events
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Mutual exclusivity
• When you and your significant other decide not to date anyone else?
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Mutual exclusivity
• When you and your significant other decide not to date anyone else?
• No.
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Mutual exclusivity
• When you and your significant other decide not to date anyone else?
• No. Not in stats class, anyways.
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Mutual exclusivity
• Mutual Exclusivity is when two events cannot both occur together
• A coin cannot be both heads and tails at the same time
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Mutual exclusivity
• Mutual Exclusivity is when two events cannot both occur together
• A coin cannot be both heads and tails at the same time
• Can you think of other examples of mutual exclusivity?
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Questions? Comments?
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Sample space
• The sample space is the set of all possible simple events
• So for a fair coin, it’s {T, H}
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What if you have a 6-sided die?
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What if you have a 6-sided die?
• Sample space = {1, 2, 3, 4, 5, 6}
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So if I have a six-sided die
• What is the probability that I roll a 6?
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So if I have a six-sided die
• What is the probability that I roll a 6?– 1/6 or 0.17
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So if I have a six-sided die
• What is the probability that I roll an odd number?
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So if I have a six-sided die
• What is the probability that I roll an odd number?– 1, 3, 5– Out of– {1, 2, 3, 4, 5, 6}
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So if I have a six-sided die
• What is the probability that I roll an odd number?– 1/2 or 0.5
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So if I have a six-sided die
• What is the probability that if I roll the die twice, I get 1 both times?
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So if I have a six-sided die
• What is the probability that if I roll the die twice, I get 1 both times?– 1/36
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Questions? Comments?
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Compound events
• Multiple simple events, one after another
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What is the probability of getting 2 heads in a row
• T, T• H, H• T, H• H, T
• 1 out of 4
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More complex combinations possible too
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Let’s look at another example
• Let’s say I flip a coin 4 times• And I get 3 heads and 1 tail– We don’t care about the order
• What is the probability of that?
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3 heads, 1 tail
• How many cases are there?
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3 heads, 1 tail
• How many cases are there?
• 16– HHHH, HHHT, HHTH, HHTT, – HTHH, HTHT, HTTH, HTTT, – THHH, THHT, THTH, THTT, – TTHH, TTHT, TTTH, TTTT,
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3 heads, 1 tail
• How many cases have 3 heads, 1 tail?
• 16– HHHH, HHHT, HHTH, HHTT, – HTHH, HTHT, HTTH, HTTT, – THHH, THHT, THTH, THTT, – TTHH, TTHT, TTTH, TTTT,
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3 heads, 1 tail
• How many cases have 3 heads, 1 tail?
• 16– HHHH, HHHT, HHTH, HHTT, – HTHH, HTHT, HTTH, HTTT, – THHH, THHT, THTH, THTT, – TTHH, TTHT, TTTH, TTTT,
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3 heads, 1 tail
• How many cases have 3 heads, 1 tail?
• 4/16 = 0.25– HHHH, HHHT, HHTH, HHTT, – HTHH, HTHT, HTTH, HTTT, – THHH, THHT, THTH, THTT, – TTHH, TTHT, TTTH, TTTT,
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This is an illustration of probability
• To quote the book
• The probability of an event A is equal to the sum of the probabilities of the simple events contained in A
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Example with 1 head out of 2 flips
• 4 cases– HH– HT (1/4)– TH (1/4)– TT
• ¼ + ¼ = ½
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Example with 3 heads out of 4 flips
• 16 cases– HHHH, HHHT, HHTH, HHTT, – HTHH, HTHT, HTTH, HTTT, – THHH, THHT, THTH, THTT, – TTHH, TTHT, TTTH, TTTT,
• 1/16 + 1/16 + 1/16 + 1/16 = 4/16 = 1/4
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Questions? Comments?
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Probabilities don’t have to be equal for all examples in the sample space
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Probabilities don’t have to be equal for all examples in the sample space
• For example, let’s say that you have made a new friend, randomly sampled from the population of New York City residents
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Probabilities don’t have to be equal for all examples in the sample space
• For example, let’s say that you have made a new friend, randomly sampled from the population of New York City residents– That’s how I make friends
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Probabilities don’t have to be equal for all examples in the sample space
• For example, let’s say that you have made a new friend, randomly sampled from the population of New York City residents– That’s how I make friends
• 56% of NYC residents take mass transit to work
• 28% of NYC residents drive to work
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Probabilities don’t have to be equal for all examples in the sample space
• For example, let’s say that you have made a new friend, randomly sampled from the population of New York City residents– That’s how I make friends
• 56% of NYC residents take mass transit to work• 28% of NYC residents drive to work
• What is the probability that your friend either takes mass transit or drives to work?
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Probabilities don’t have to be equal for all examples in the sample space
• For example, let’s say that you have made a new friend, randomly sampled from the population of New York City residents– That’s how I make friends
• 56% of NYC residents take mass transit to work• 28% of NYC residents drive to work
• What is the probability that your friend either takes mass transit or drives to work?– 84%
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Questions? Comments?
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If we have time
• Left-over from previous class
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If we have time
• Demo finding A and B for Y = A + Bx
• In Excel
• Using Sum of Squared Residuals– Residual = Difference Between Predicted Y and
Actual Y
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Questions? Comments?
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Upcoming Classes
• 2/9 No class• 2/11 Permutations, Combinations, Unions, and
Complements– Ch. 4.4– HW2 due
• 2/13 Independence and Conditional Probability– Ch. 4.5, 4.6
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Homework 2
• Due in 7 days• In the ASSISTments system
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Questions? Comments?