more of david palay’s slides
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More of David Palay’s Slides. Conditional Probability. Given that you had a quiz on Tuesday, what is the Probability that you have homework tonight…. Example:. - PowerPoint PPT PresentationTRANSCRIPT
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More of David Palay’sSlides
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Conditional Probability
Given that you had a quiz on Tuesday, what is the Probability that you have
homework tonight…
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Example:
Suppose one employee is selected from a group of random employees. We’re interested if this employee has used the proper cover sheet for his TPS report (since there’s a new cover sheet and we’re not certain that this employee has read the memo). We could ask:• What is We could ALSO ask,• What is
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P(improper TPS∨Male)• This is the probability of the employee’s TPS
report having the wrong coversheet, GIVEN that the employee is male.
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Conditional Probability
• Written as • Read asProbability of event B, given event A has already happened.
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Our Company
• There are 15 people in the small company.• 9 are male• 5 females filed improper TPS reports• 8 employees filed improper TPS reports.
• From this, we can figure out the probability of having an improper TPS report given that the employee is male.
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Method 1
• Create a table:
• Use table to calculate probability
Male Female Total
Proper TPS
Improper TPS
Total
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Example
• When rolling 2d6, what is the probability that the sum is a 7, GIVEN THAT ONE OF THE DICE IS A 1.
{1,1} {1,2} {1,3} {1,4} {1,5} {1,6}{2,1} {2,2} {2,3} {2,4} {2,5} {2,6}{3,1} {3,2} {3,3} {3,4} {3,5} {3,6}{4,1} {4,2} {4,3} {4,4} {4,5} {4,6}{5,1} {5,2} {5,3} {5,4} {5,5} {5,6}{6,1} {6,2} {6,3} {6,4} {6,5} {6,6}
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Method 2: Calculating
P (B|A )= P (A ∩B )P (A )
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TPS Reports
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Dependent and Independent events
• Dependent events– 2 events where the outcome of one is influenced
by the outcome of the other. • Independent events– 2 events who’s outcomes are completely
unchanged based on the outcomes of each other.
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Dependent and Independent events
• If • Or if • Then the events are independent!
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Tossing two Coins
• If A is getting heads on the first toss, and• If B is getting heads on the second toss,Are A and B dependent or independent?
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Tossing two Coins
• Find • Find
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Tossing two Coins
• Event A: getting Heads on first toss• Event B: getting Heads on second toss• Find
• Find
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A useful result!
• Take the mathematical equation of conditional probability:
Multiply both sides by We can calculate with just and . This is the MULTIPLICATION RULE for intersections.
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Even more useful!
• If we think about independent events for a minute, we can recall that for two events A and B that are independent:
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But that means…
In our Multiplication Rule for Intersections
If we are looking at ONLY independent events where
Then, the Multiplication Rule results in
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This works for the probability of more than 2 events!
• If A, B, C, D, … are all INDEPENDENT events (coin flips, single die rolls, etc.), then
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Ok, take a second…. Breath…
• We just covered a lot with very little explanation. So, we’re going to do some examples.
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Dependent vs. Independent
Determine whether the events are independent or dependent
1) A: a coin comes up heads B: 1d6 comes up even
2) A: 1d6 is thrown and comes up evenB: a second d6 is thrown and the sum of the two is greater than 4
3) A: An ace is drawn from a standard deck of cardsB: A second ace is drawn from the same deck
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Uh-oh…. Wait a sec.
A: An ace is drawn from a standard deck of cardsB: A second ace is drawn from the same deck
We can come up with two different answers. It depends on if we put the first card BACK or NOT!
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Replacement
• We talked about it with combinations & permutations, and now we need to incorporate it into independent & dependent events.
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Replacement
Namely, if we have replacement, does it make it dependent or independent?
Take our deck of cards example.
A: An ace is drawn from a standard deck of cardsB: A second ace is drawn from the same deck
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Replacement
If we put the first card back into the deck (and re-shuffle) then we have an independent event. The first card has no impact on what the second card could be.
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Replacement
Alternatively, if we leave the first card out, then the probabilities for the second card have changed.
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One more thing
• Find
• Find