Question

Suppose that you start with a normal deck of 52 cards and remove everything except the twos, threes, and fours. So you now have a deck of twelve cards consisting of four twos, four threes, and four fours. A card is drawn at random and replaced then another card is drawn at random and replaced. Make a probability distribution for the sum of the two numbers that are drawn. Hint: List all the possible sums of two cards given 4 twos, four 3’s and four 4’s. Then count how many outcomes correspond to each sum!

Answer 1

It is given that from a normal deck of 52 cards we will remove everything except the twos, threes and fours. So, now we have a deck of 12 cards consisting of four twos, four threes and four fours. It is given that a card is drawn at random and replaced, then another card is drawn at random and replaced.

As each card that is drawn is replaced before drawing the next card, the probability of each number on each draw = 4/12 = 1/3 .

Now, we have to make a probability distribution for the sum of the two numbers that are drawn.

So,

P(4) = P(2 and 2) = (1/3)(1/3) = 1/9

P(5) = P(2 and 3) or P(3 and 2) = (1/3)(1/3) + (1/3)(1/3) = 1/9 + 1/9 = 2/9

P(6) = P(2 and 4) or P(3 and 3) or P(4 and 2) = (1/3)(1/3) + (1/3)(1/3) + (1/3)(1/3) = 1/9 + 1/9 + 1/9 = 3/9

P(7) = P(3 and 4) or P(4 and 3) = (1/3)(1/3) + (1/3)(1/3) = 1/9 + 1/9 = 2/9

P(8) = P(4 and 4) = (1/3)(1/3) = 1/9

Thus, given above are the required answers .