2019 Worksheet-7-Conditional Probability PDF

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Summary

Worksheet for 2019 winter quarter which begun January 4, 2019...

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Math 130

Worksheet 7

Conditional Probability

Name ___________________________________ Basic Concepts: Conditional Probability: The probability that event A occurs, given that event B occurs, is P[A  B ] P[A | B ]  (The probability of A given B) P[B ] Independent Events: The events A and B are independent if P[ A | B] P[ A ]

(Definition of independent events)

An equivalent condition for independence is P[ A  B] P[ A]P[ B ]

(Equivalent condition for independence)

1. An experiment consists of tossing a fair coin and rolling a fair die. Thus, a typical outcome is H 3 . Define two events as follows: A is the event that the coin was a H. B is the event that the die was a 4. a. How many outcomes are there in the sample space for this experiment? Die H T

1 H1 T1

2 H2 T2

3 H3 T3

4 H4 T4

5 H5 T5

6 H6 T6

2 outcomes for coin and 6 outcomes for dice n(S) = 2*6 =12 b. How many outcomes are in A? What is the probability of A? Outcomes for head = 6 Probability P(A) = 6/12 = 1/2 c. How many outcomes are in B? What is the probability of B? Outcomes for 3 = 2 Probability = 2/12 =1/6 d. How many outcomes are in A  B ? What is the probability of A  B ?

Only 1 outcome Sample space = 12 Probability = 1/12 e. Find the conditional probability P[ A | B] . P[A  B ] P[A | B ]  P[ B ] P[A|B] = 1/2 f. Are the events A and B independent? Explain with probability equation. P[A|B] = P[A] = ½ hence independent 2. A fair die is rolled twice and the following events are defined: A is the event that the sum of the two rolls is odd. B is the event that the sum of the two rolls is greater than 8. a. Find P[ A ] . 2+4+6+4+2 = 18 Sample space = 36 P[ A ] =18/36 =1/2 b. Find P[ B] 4+3+2+1 = 10 P[ B] =10/36 = 5/18 c. Find P[ A  B] by finding how many outcomes in A  B first. P[ A  B] =6/36 = 1/6

d. Find P[ A | B] by using the definition of conditional probability. P[ A | B] = P[ A  B ] / P[ B] P[ A | B] = 6/ 10 = 5/3

e. Are the events A and B independent? Explain with a probability equation. No they are not independent because P[A|B] is not equal P[A]

3. Suppose S is a sample space and A and B are two events which are mutually exclusive. Are these events independent? Explain with a probability equation.

4. Consider the experiment of dealing a 5-card hand from a deck. Define events as follows: A is the event consisting of all hands that are flushes. (Including royal flush, straight flush; a hand of cards of the same suit) B is the event consisting of all hands containing at least one 7. a. Find P[ A ] .

b. Find P[ B] .

c. Find P[ A  B] by counting how many 5-card hands in A  B first. (That is however many flushes including a 7 card.)

d. Are the events A and B independent? Explain with a probability equation....