The score, X, for a biased spinner is given by the probability distribution | x | 0 | 3 | 6 | |----|----|----|----| | P(X = x) | 1/12 | 2/3 | 1/4 | Find (a) E(X) (b) Var(X) A biased coin has one face labelled 2 and the other face labelled 5 - Edexcel - A-Level Maths: Statistics - Question 6 - 2017 - Paper 1

To find the variance, Var(X), we first need E(X^2):

$E(X^2) = ext{Sum of }(x^2 \times P(X = x))$

Calculating each term:

• For x = 0: $0^2 \times \frac{1}{12} = 0$
• For x = 3: $3^2 \times \frac{2}{3} = 6$
• For x = 6: $6^2 \times \frac{1}{4} = 9$$

Thus,

$E(X^2) = 0 + 6 + 9 = 15$

Now, we can calculate the variance:

$Var(X) = E(X^2) - (E(X))^2 = 15 - (3.5)^2 = 15 - 12.25 = 2.75$

The expected value of Y is given by:

$E(Y) = 2 \times (1 - p) + 5 \times p = 3$

Expanding gives:

$2 - 2p + 5p = 3$

Combining terms:

$2 + 3p = 3$

Thus,

$3p = 1 \Rightarrow p = \frac{1}{3}$

From the calculations:

• P(Y = 2) = 2/3
• P(Y = 5) = 1/3

Thus, the probability distribution is:

Sam's score S can equal 30 if:

• If X = 6, then S = 6Y. To satisfy S = 30, Y must equal 5:

$P(S = 30) = P(X = 6) \times P(Y = 5)$

Using the probabilities:

$P(X = 6) = \frac{1}{4}, P(Y = 5) = \frac{1}{3}$

Thus,

$P(S = 30) = \frac{1}{4} \times \frac{1}{3} = \frac{1}{12}$

The score S can take the following values:

• S = 0 when X = 0 and Y = 0
• S = 2 when X = 0 and Y = 2
• S = 36 when X = 6 and Y = 6, etc.

We sum these probabilities to find the possible values:

For the expected value E(S):

$E(S) = \sum_{i}(s_i \times P(S = s_i))$

You compute this based on the earlier results of the distribution:

$E(S) = 0 + (2 \times P(S=2)) + (6 \times P(S=6)) + (12 \times P(S=12)) + (15 \times P(S=15)) + (30 \times P(S=30))$

Calculating these terms using the probabilities obtained:

This is equal to: 11.416

Charlotte uses $X^2$ as her score, which is always greater or equal to Sam's score when X is non-zero. The expected value of $E(X^2)$ will generally exceed that of $E(S)$ due to the nature of squaring larger values, thus leading to higher total scores. Therefore, Charlotte should achieve the higher score.