Question
Gustav plays a game in which he first tosses an unbiased coin and then rolls an unbiased six-sided die.
- If the coin shows tails, the score on the die is Gustav’s final number of points.
- If the coin shows heads, one is added to the score on the die for Gustav’s final number of points.
(a) Find the probability that Gustav’s final number of points is 7.
(b) Find the probability distribution for Gustav’s final number of points.
(c) Calculate the expected value of Gustav’s final number of points.
▶️ Answer/Explanation
Detailed Solution
(a) Probability of Gustav Scoring 7
To score 7, Gustav must get:
- A head on the coin toss (\(\frac{1}{2}\))
- A 6 on the die (\(\frac{1}{6}\))
Calculating the combined probability:
\[ P(7) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} = 0.0833 \]
(b) Probability Distribution for Gustav’s Final Number of Points
Since each point can be achieved in two ways:
- For point \( x \), it occurs if:
- The coin shows heads, and the die rolls \( x – 1 \).
- The coin shows tails, and the die rolls \( x \).
- Each outcome has a probability of \( \frac{1}{12} \).
Thus, the probability for each score is calculated as:
\[ P(X = x) = 2 \times \left(\frac{1}{6} \times \frac{1}{2} \right) = \frac{2}{12} = \frac{1}{6} \]
The probability table is:
(c) Expected Value of Gustav’s Final Number of Points
Using the expectation formula:
\[ E(X) = \sum x P(X = x) \]
\[ E(X) = (1 \times \frac{1}{12}) + (2 \times \frac{1}{6}) + (3 \times \frac{1}{6}) + (4 \times \frac{1}{6}) + (5 \times \frac{1}{6}) + (6 \times \frac{1}{6}) + (7 \times \frac{1}{12}) \]
Since the probabilities are symmetric, the expected value simplifies to:
\[ E(X) = 3.5 + 0.5 = 4 \]
Conclusion: The expected number of points Gustav scores is 4.
……………………………Markscheme……………………………….
(a) Probability of scoring 7: \( \frac{1}{12} = 0.0833 \)
(b) Probability distribution: Shown in the table.
(c) Expected value: 4