In statistics class 78% students study very hard, 90% of those are female studen

Question

In statistics class 78% students study very hard, 90% of those are female students, while 93% male students do not work so hard. Draw a tree propability diagram Find Pr(randomly selected female) Find Pr( randomly selected male) Find Pr (male/study hard) Find Pr(study hard/male).

From 6 women and 5 men a committee of 7 persons to be formed.

a. How many different committees can be formed?

b. How many committees contain 3 women and 4 men can be formed?

c. What is the probability that the committee has at most 2 men?

d. What is the probability that the committee has at least 4 women? e. What is the probability that the committee has exactly 6 men?

3. (A four-digit PIN number cannot start with a 0.

a. How many variations are possible?

b. How many variations are possible, if all digits are the same?

c. What is the probability that the PIN has all different digits?

d. What is the probability that the PIN has a 0 in the end?

4. You are dealt a bridge hand 13 cards) from a standard deck. Find the probability that:

a. Every card is a spade.

b. You picked 3 Aces, 2 Kings, 3 10s, 4 6s, and 1 Queen.

c. You picked 4 Diamonds, 2 Hearts, 3 Clubs, 4 Spades.

d. You are holding nothing but face cards and Aces.

Explanation / Answer

In statistics class 78% students study very hard, 90% of those are female students, while 93% male students do not work so hard. Draw a tree propability diagram Find Pr(randomly selected female) Find Pr( randomly selected male) Find Pr (male/study hard) Find Pr(study hard/male).

Here we have

P(study hard) = 0.78, P(female | study hard ) = 0.90, P(male | study not hard ) = 0.93

By the complement rule

P(study not hard) = 1 - P(study hard) = 0.22

P(female | study not hard ) =1- P(male | study not hard ) = 1 - 0.93 = 0.07

By the law of total probability, the probability of selecting a female is

P(randomly selected female) = P(female | study not hard )P(study not hard) + P(female | study hard )P(study hard) = 0.07*0.22 + 0.90* 0.78 = 0.7174

By the complement rule, we have

P(randomly selected male)=1 -P(randomly selected female) = 1- 0.7174 = 0.2826

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By the complement rule

Pr (male| study hard) =1- P(female | study hard ) = 1-0.90=0.10

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Now

Pr(study hard | male) = [P(male |study hard)P(study hard) ] / P(male) = [0.10*0.78] / 0.2826 = 0.2760

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