A fair blue die has faces numbered 1, 1, 3, 3 and 5 - Edexcel - A-Level Maths Statistics - Question 6 - 2013 - Paper 1

This is known as the Discrete Uniform Distribution.

The expected value E(B) is calculated as follows:

E(B) = rac{1}{5}(1) + rac{2}{5}(3) + rac{1}{5}(5) = rac{1 + 6 + 5}{5} = rac{12}{5} = 2.4

To find E(R), we calculate:

E(R) = 2 imes rac{2}{3} + 4 imes rac{1}{6} + 6 imes rac{1}{6}

Calculating this gives:

E(R) = rac{4}{3} + rac{4}{6} + rac{6}{6} = rac{4}{3} + rac{10}{6} = rac{4}{3} + rac{5}{3} = rac{9}{3} = 3

To find Var(R), we first need E(R²):

E(R^2) = 2^2 imes rac{2}{3} + 4^2 imes rac{1}{6} + 6^2 imes rac{1}{6}

Calculating gives:

E(R^2) = rac{4}{3} + rac{16}{6} + rac{36}{6} = rac{4}{3} + rac{52}{6} = rac{4}{3} + rac{26}{3} = rac{30}{3} = 10

Now we find the variance:

$Var(R) = E(R^2) - (E(R))^2 = 10 - 3^2 = 10 - 9 = 1$

When the coin shows 2, Avisha will choose the blue die, as the scores (1, 3, 5) are more likely to be greater than 2. The probability that she wins when rolling the blue die:

P(Avisha ext{ wins | coin = 2}) = P(B > 2) = P(B = 3) + P(B = 5) = rac{2}{5} + rac{1}{5} = rac{3}{5}

When the coin shows 5, Avisha will choose the red die, since it can either show 6 or less. The probability that she wins:

$P(Avisha ext{ wins | coin = 5}) = P(R > 5) = 0$

Now, considering the outcome of the coin:

$P(Avisha ext{ wins}) = P( ext{Coin = 2}) imes P(Avisha ext{ wins | coin = 2}) + P( ext{Coin = 5}) imes P(Avisha ext{ wins | coin = 5})$

Thus,

P(Avisha ext{ wins}) = rac{1}{2} imes rac{3}{5} + rac{1}{2} imes 0 = rac{3}{10}.