James is playing a mathematical game on his computer - AQA - A-Level Maths Pure - Question 17 - 2021 - Paper 3

To test James's claim, we start by defining the hypotheses:

• Null hypothesis ($H_0$): $p = 0.6$ (the probability of winning has not increased).
• Alternative hypothesis ($H_1$): $p > 0.6$ (the probability of winning has increased).

Using the binomial distribution, we can calculate:

1. The probability of winning exactly 12 games out of 15 when $p = 0.6$: $P(X = 12) = \binom{15}{12} (0.6)^{12} (0.4)^{3}$ After performing the calculations, we find: $P(X \geq 12)$ using normal approximation or binomial tables.

2. The critical region is determined for a 5% significance level, where we find the critical value such that: $P(X \geq k) < 0.05$ for the sample size of 15.

3. If the calculated $P(X \geq 12)$ is greater than the significance level, we fail to reject $H_0$; otherwise, we reject $H_0$ and accept $H_1$.

In conclusion, if $P(X \geq 12)$ is not significant, then there is insufficient evidence to suggest that James's claim regarding his probability of winning the game has increased.