1 A website simulates the outcome of throwing four fair dice. Ten thousand people take part in a challenge using the website in which they have one attempt at getting four sixes in the four throws of the dice. The number of people who succeed in getting four sixes is denoted by the random variable \(X\).
- Show that, for each person, the probability that the person gets four sixes is equal to \(\frac { 1 } { 1296 }\).
- Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
- Use a Poisson distribution to calculate each of the following probabilities.
- \(\mathrm { P } ( X = 10 )\)
- \(\mathrm { P } ( X > 10 )\)
- In another challenge on the website, 50 people are each given 20 independent attempts to try to get four sixes as often as they can.
Determine the probability that no more than 2 people succeed in getting four sixes at least once in their 20 attempts.