Patrick is practising his skateboarding skills - AQA - A-Level Maths Pure - Question 13 - 2019 - Paper 3

The variance of a binomial distribution can be calculated using the formula:

Substituting the values:

• $n = 30$
• $p = 0.2$

We have:

Calculating:

Therefore, the variance is 4.8.

To find the probability of exactly $k$ successes in a binomial distribution, we use:

For this problem, with $n = 30$, $k = 10$, and $p = 0.2$, we have:

Calculating this yields:

$P(X = 10) ext{ = approximately } 0.0355$

Thus, the probability of falling off exactly 10 times is approximately 0.0355.

To find the probability of falling off 5 or more times, we can derive it from the cumulative probability:

$P(X ext{ } ext{≥ 5}) = 1 - P(X < 5)$

which is equivalent to:

$P(X ext{ } ext{≥ 5}) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)]$

Using the binomial formula to calculate these probabilities:

After calculating, we find:

$P(X ext{ } ext{≥ 5}) ext{ = approximately } 0.745$

Therefore, the probability of falling off 5 or more times is approximately 0.745.

The probability he falls off at least 5 times on any single day is already calculated as:

$P(X ext{ } ext{≥ 5}) ext{ = } 0.745$

For 5 consecutive days, assuming that each day's probability is independent, we raise this probability to the power of 5:

$P( ext{At least 5 times on 5 days}) = (0.745)^5$

Calculating this gives:

$(0.745)^5 ext{ = approximately } 0.229$

Thus, the probability of falling off at least 5 times on each of the 5 days is approximately 0.229.

Using a constant probability of 0.2 assumes that Patrick's skill level and ability remain unchanged throughout the 5 days. However, as he practices, he is likely to improve, and his likelihood of falling off may decrease. Factors such as fatigue or external conditions could also affect his performance, leading to variability in the probability of falling off on different days.