After practising the game, James claims that he has increased his probability of winning the game - AQA - A-Level Maths Mechanics - Question 17 - 2021 - Paper 3

Using the binomial distribution:

• Let $X$ be the number of games won.
• We need to calculate $P(X \geq 12)$, which is equivalent to $1 - P(X \leq 11)$.

Using the binomial formula: $P(X \leq 11) = \sum_{k=0}^{11} \binom{15}{k} (0.6)^k (0.4)^{15-k}$

From calculations, we find:

$P(X \leq 11) = 0.0909$ Thus, $P(X \geq 12) = 1 - P(X \leq 11) = 1 - 0.0909 = 0.9091$

Since $0.9091 > 0.05$, we do not reject $H_0$, indicating there is insufficient evidence to suggest that the probability of winning the game has increased.

In conclusion, based on the sample of 15 games, the data does not provide sufficient evidence to support the claim that James’s probability of winning has increased.