Studocu Worksheet-7-Conditional Probability PDF

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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 ]  P[B ] (The probability of A given 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? 2 H ∨T × 6 die=12 outcomes

b. How many outcomes are in A? What is the probability of A? P ( A )=

6 1 = 12 2

c. How many outcomes are in B? What is the probability of B? 2 1 P ( B) = = 12 6 d. How many outcomes are in A  B ? What is the probability of A  B ? 4H can occur only once in 12 possibilities P( A  B ) = 1/12

e. Find the conditional probability P[ A | B] .

P[A | B ]  P [ A|B ]=

P[A  B ] P [B ]

1 6 1 × = 12 1 2

f. Are the events A and B independent? Explain with probability equation. Yes they are independent because P[A|B] = P[A]. 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 ] . P ( A )=

18 1 = 36 2

b. Find P[ B] P ( B) =

10 5 = 36 18

c. Find P[ A  B] by finding how many outcomes in A  B first. ¿

4 1 = 36 9

d. Find P[ A | B] by using the definition of conditional probability. 1 18 2 ¿= ❑ = × = ❑ 9 5 5 e. Are the events A and B independent? Explain with a probability equation. No, they are not because P[A|B] is not equal to 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. Mutually exclusive events cannot be independent because: In mutually exclusive events P(A∣B) = 0 and P(A∪B) = P(A) + P (B) but in independent events P(A∣B) = P(A) (and P(A∣B) cannot be zero) and P(A∪B) = P(A) + P (B) - P(A) P(B).

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 ] .

( 525)=2598960 Sample ( A ) = (4 ) × ( 13 )=5148 1 5 sample space=

P ( A )=

5148 =0.00198 2598960

b. Find P[ B] .

( 525)=2598960 Sample ( B ) =( 48 ) =1712309 5 sample space= c

P ( B ) =1−

1712309 =0.000762 2598960

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.)

( )

sample space= 52 =2598960 5

(41 ) × (124) =1980

N ( A ∩ B) =

1980 =0.000762 2598960 d. Are the events A and B independent? Explain with a probability equation. P ( A n B )=P ( A ) × P ( B )=0.00198 × 0.391158= 0.00077 Events A and B are independent because P ( A n B ) ¿ part C is equal¿ P ( A n B ) ∨P ( A ) × P ( B ) above

P ( A ∩ B)=...