7.2: Drawing two cards (or three cards) out of a standard deck of 52 cards witho

Question

7.2: Drawing two cards (or three cards) out of a standard deck of 52 cards without replacement. (a) What is the probability of drawing exactly a face card? (b) What is the probability of drawing no face card? (c) What is the probability of drawing at least one face card? (Hint: use combination formula)

8.1: An experiment consists of tossing a coin three times. Let Y be the number of times that the head turns up. (a) Is Y a random variable? (b) What are those possible values that Y can take on? (c) What is the probability that Y can take on the value of one?

8.2: If you randomly (without looking) pick two marbles with replacement out of an urn that contains 8 red and 10 blue marbles, let Y be the number of red marbles selected. (a) Is Y a random variable? (b) What are those possible values that Y can assume? (c) What is the probability that Y can take on the value of one?

8.3: You and a friend play a game where you randomly select one card out of an ordinary deck of 52 cards. If the card selected is a spade, you will win $3; otherwise, you lose $1 (win (-$1)). Give the probability distribution for your winning amount, Y, on a single play of this game.

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Explanation / Answer

7.2 )

Drawing two cards (or three cards) out of a standard deck of 52 cards without replacement. (a) What is the probability of drawing exactly a face card?

there are 12 face cards in a deck , so 1 face card can be drawn out of 12 in 12C1 ways and the remaining 1 card cane be drawn in (52-12) = 40C1 ways out of total ways of 52C2 ways , so probability becomes

12C1*40C1/52C2 = 12*40/1326 = 0.361

(b) What is the probability of drawing no face card?

Again total ways = 52C2

first card a face = 12C1

second card can be drawn in 11C1 ways , as one non face card is drawn without replacement

so 12C1*11C1/52C2 = 12*11/1326 = 0.099 , so final probability is 1 - 0.099 = 0.901

(c) What is the probability of drawing at least one face card?

again total ways = 52C2

first card a face = 12C1

second card a face is 11C1 , as one is drwan without replacement

11C1*12C1/52C2 = 0.099

or

1st face and second no face = 12C1*40C1/52C2 = 0.36

so finally 2*0.36+0.099 = 0.819 ,(we multiply by 2 as first can be no face and second can be a face , which is again2C1*40C1/52C2 )

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