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Nov 2007 p6 q6
3339
On any occasion when a particular gymnast performs a certain routine, the probability that she will perform it correctly is 0.65, independently of all other occasions.
On one day she performs the routine 50 times. Use a suitable approximation to estimate the probability that she will perform the routine correctly on fewer than 29 occasions.
Solution
The number of times the gymnast performs the routine correctly follows a binomial distribution with parameters \(n = 50\) and \(p = 0.65\).
We approximate this binomial distribution with a normal distribution. The mean \(\mu\) and variance \(\sigma^2\) of the binomial distribution are given by:
\(\mu = n \times p = 50 \times 0.65 = 32.5\)
\(\sigma^2 = n \times p \times (1-p) = 50 \times 0.65 \times 0.35 = 11.375\)
We use a continuity correction to find the probability of fewer than 29 occasions:
