ProbabilityBy Mr.Golam

Robbani Ahmed

What are the Chances?

A probability is something that can happen in more than one way.

The outcomes of an experiment are the ways it can happen.

The event is the particular outcome we are looking for.

6 10 52

?H HT TH T T H

The Probability of an Event P(Event) = the number of ways the event can happen the number of all possible outcomes

P(H+T) = 2 = 1 4 2

__ __

BA

BAA

B14 12

35

2514

a) b)

P(red)

P(blue)

P(yellow or blue)1

5

What is the probability of getting 4 fives?P(4 fives) =

4 52

CARD

What is the probability of getting an even number?

213

DICE

P(even) =

What is the probability of rolling two coins and getting H first and then T?

P(H & then T) =

COINS

14

Random Experiment…

•…a random experiment is an action or process that leads to one of several possible outcomes. For example:

Experiment Outcomes

Tossing a coin Heads, Tails

Exam Marks Numbers: 0, 1, 2, ..., 100

Throwing a die 1,2,3,4,5,6

Classical Approach…

•If an experiment has n possible outcomes [all equally likely to occur], this method would assign a probability of 1/n to each outcome.

•Experiment: Rolling a die•Sample Space: S = {1, 2, 3, 4, 5, 6}•Probabilities: Each sample point has a 1/6 chance of occurring.

Classical Approach

•Experiment: Throwing two dies simultaneously•Sample Space: S = {2, 3, …, 12}•Probability Examples:•P(2) = 1/36

•P(7) = 6/36•P(10) = 3/36•

1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

What are the underlying, unstated assumptions??

Three coins are tossed. Find the probability of: 1. 3 heads 2. At least 2 heads. 3. At most 2 heads. At least 2 heads means minimum 2 heads i.e. 2 or more heads i.e. 3 .At most means maximum 2 heads i.e. up to 2 heads, i.e. less than or equal to 2 heads or no head.No. of favorable outs comes=(HHH),(HTH),(HTT),(THH),(THT),(TTT),(HHT)(TTH)=81.3 Head=3/82.At least 2 head=4/8=1/23.At most 2 head=7/8