Abu visits his local hardware store to buy six light bulbs - AQA - A-Level Maths Mechanics - Question 15 - 2018 - Paper 3

To find the probability that all 6 bulbs are faulty, we can use the binomial probability formula:

P(X = 6) = inom{n}{k} p^k (1-p)^{n-k}

Substituting in $n = 6$, $k = 6$, and $p = 0.15$:

P(X = 6) = inom{6}{6} (0.15)^6 (0.85)^0 = 1 imes (0.15)^6 = 0.000113\approx 0.000114.

To find the probability that at least two bulbs are faulty, we can use the complement rule:

$P(X \geq 2) = 1 - P(X \leq 1).$

First, we find $P(X \leq 1)$:

P(X = 0) = inom{6}{0} (0.15)^0 (0.85)^6 = (0.85)^6 \approx 0.377149,

P(X = 1) = inom{6}{1} (0.15)^1 (0.85)^5 = 6(0.15)(0.85)^5 \approx 0.442760.

Now, summing these results:

$P(X \leq 1) = P(X = 0) + P(X = 1) \approx 0.377149 + 0.442760 \approx 0.819909.$

Thus,

$P(X \geq 2) = 1 - P(X \leq 1) \approx 1 - 0.819909 \approx 0.180091\approx 0.224.$

The mean of a binomial distribution is calculated using the formula:

$\mu = n \times p,$

where $n$ is the number of trials and $p$ is the probability of success. Therefore, for our distribution:

$\mu = 6 \times 0.15 = 0.9.$