Question

you are playing a game with a deck of cards that only contains cards labeled with a 2, 3, 4, or 5. each value appears on the same number of cards. as part of this game, you draw one card from the deck, replace it and shuffle the deck, then draw a second card. your score is the sum of the numbers on those cards. fill in the probability distribution below. please enter your answers as fractions. score 4 5 6 7 8 9 10 probability

Explanation:

Step1: Calculate total number of outcomes

Since there are 4 card - values and we draw 2 cards with replacement, the total number of outcomes is \(4\times4 = 16\).

Step2: Find number of ways to get score 4

The only way to get a sum of 4 is \((2,2)\), so there is 1 way. Probability \(P(4)=\frac{1}{16}\).

Step3: Find number of ways to get score 5

The ways are \((2,3)\) and \((3,2)\), so there are 2 ways. Probability \(P(5)=\frac{2}{16}=\frac{1}{8}\).

Step4: Find number of ways to get score 6

The ways are \((2,4)\), \((4,2)\), \((3,3)\), so there are 3 ways. Probability \(P(6)=\frac{3}{16}\).

Step5: Find number of ways to get score 7

The ways are \((2,5)\), \((5,2)\), \((3,4)\), \((4,3)\), so there are 4 ways. Probability \(P(7)=\frac{4}{16}=\frac{1}{4}\).

Step6: Find number of ways to get score 8

The ways are \((3,5)\), \((5,3)\), \((4,4)\), so there are 3 ways. Probability \(P(8)=\frac{3}{16}\).

Step7: Find number of ways to get score 9

The ways are \((4,5)\), \((5,4)\), so there are 2 ways. Probability \(P(9)=\frac{2}{16}=\frac{1}{8}\).

Step8: Find number of ways to get score 10

The only way is \((5,5)\), so there is 1 way. Probability \(P(10)=\frac{1}{16}\).

Answer:

Score	4	5	6	7	8	9	10