Question

Problem 1 (3 + 3 + 3 = 9) Suppose you draw two cards from a deck of 52 cards without replacement. 1) What’s the probability that both of the cards are hearts? 2) What’s the probability that exactly one of the cards are hearts? 3) What’s the probability that none of the cards are hearts?

Problem 2 (4) A factory produces 100 unit of a certain product and 5 of them are defective. If 3 units are picked at random then what is the probability that none of them are defective?

Problem 3 (3+4=7) There are 3 bags each containing 100 marbles. Bag 1 has 75 red and 25 blue marbles. Bag 2 has 60 red and 40 blue marbles. Bag 3 has 45 red and 55 blue marbles. Now a bag is chosen at random and a marble is also picked at random. 1) What is the probability that the marble is blue? 2) What happens when the first bag is chosen with probability 0.5 and other bags with equal probability each?

Probem 4 (3+3+4=10) Before each class, I either drink a cup of coffee, a cup of tea, or a cup of water. The probability of coffee is 0.7, the probability of tea is 0.2, and the probability of water is 0.1. If I drink coffee, the probability that the lecture ends early is 0.3. If I drink tea, the probability that the lecture ends early is 0.2. If I drink water, the lecture never ends early. 1) What’s the probability that I drink tea and finish the lecture early? 2) What’s the probability that I finish the lecture early? 3) Given the lecture finishes early, what’s the probability I drank coffee?

Problem 5 (4+4+4=12) We roll two fair 6-sided dice. Each one of the 36 possible outcomes is assumed to be equally likely. 1) Find the probability that doubles were rolled. 2) Given that the roll resulted in a sum of 4 or less, find the conditional probability that doubles were rolled. 3) Given that the two dice land on different numbers, find the conditional probability that at least one die is a 1. Problem 6 (8) For any events A, B, and C, prove the following equality: P(B|A) P(C|A) = P(B|A ∩ C) P(C|A ∩ B)

Answer 1

Dear student we can provide you with the solution of one question and 4 sub-question at a time, please repost for the rest.

Problem 1)

1) Total number of ways to draw 2 cards from a deck of 54 cards is

There is a total of 13 cards, the total number of ways to draw 2 hearts is

The probability that both the cards is heart is

2) If the first card drawn is heart than the other one is no heart. Similarly if the first card drawn is no heart than the other one is heart.

in-deck of 52 there is 13 hearts and 39 non-heart cards.

the probability that the first card is drawn is the heart and the other one is not heart is ( remember that the card is drawn without replacement hence there is 51 in the denominator of the second fraction, as a heart card is drawn as first card))

Similarly, the probability that the first card is drawn is the no heart and the other one is heart is  

hence the probability of drawing only one heart card is

3) the total number of ways to draw two non-hearts cards is

The probability that none of the cards are hearts is