you are playing a game with a deck of cards that only contains cards labeled with a 2, 3, 4, or 5. each…
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
Answer
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 |
|---|---|---|---|---|---|---|---|
| Probability | (\frac{1}{16}) | (\frac{1}{8}) | (\frac{3}{16}) | (\frac{1}{4}) | (\frac{3}{16}) | (\frac{1}{8}) | (\frac{1}{16}) |