<p>Twelve fair dice are thrown and  denotes the number of sixes obtained. If twelve dice are thrown $65$ times, estimate the probability that the average value of  is less than $1.8$.</p>

Twelve fair dice are thrown and  denotes the number of sixes obtained. If twelve dice are thrown $65$ times, estimate the probability that the average value of  is less than $1.8$.

Central Limit Theorem

Sampling

H2 Math

H1 Math

Worked Example 5

$X \sim B\left(12, \displaystyle\frac{1}{6}\right)$

$X_1 \sim B\left(12, \displaystyle\frac{1}{6}\right)$, $X_2 \sim B\left(12, \displaystyle\frac{1}{6}\right)$, ..., $X_{65} \sim B\left(12, \displaystyle\frac{1}{6}\right)$

Since $n = 65$ is large, $E(X) = 12 \times \displaystyle\frac{1}{6} = 2$

$\text{Var}(X) = 12 \times \displaystyle\frac{1}{6} \times \frac{5}{6} = \frac{5}{3}$

By Central Limit Theorem,
$\bar{X} \sim N\left(E(X), \displaystyle\frac{\text{Var}(X)}{65}\right)$ approximately

$\bar{X} \sim N\left(2, \displaystyle\frac{5/3}{65}\right)$

Required Prob $= P(\bar{X} < 1.8) = 0.106$ (3 s.f.)

Worked Example 5

Worked Example 5

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