This problem follows a binomial probability distribution since each ticket has two possible outcomes: winning (success) or losing (failure). The probability of success is 12% (0.12), and Neil buys 10 tickets.

The binomial probability formula is:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

where:

• $n = 10$ (total tickets bought)
• $k = 3$ (desired winning tickets)
• $p = 0.12$ (probability of winning per ticket)
• $\binom{n}{k}$ is the binomial coefficient:
  $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

In Excel, you can calculate this using the BINOM.DIST function:

excelCopy code
=BINOM.DIST(3, 10, 0.12, FALSE)

Steps:

1. Open Excel.
2. Type the formula =BINOM.DIST(3, 10, 0.12, FALSE).
3. Press Enter.
4. Excel will return the probability.

Rounded Answer:

After evaluating, the probability is 0.230 (rounded to three decimal places).

Would you like a breakdown of the calculations or further clarifications?

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Related Questions:

1. What is the probability that at most three of Neil's tickets have prizes?
2. What is the probability that at least one of Neil’s tickets has a prize?
3. How would the probability change if the winning percentage increased to 15%?
4. What is the expected number of winning tickets in Neil's purchase?
5. How would the probability distribution look if he bought 20 tickets instead?

🎯 Tip:

For a binomial probability distribution, the mean is given by $\mu = np$ and the standard deviation by $\sigma = \sqrt{np(1-p)}$. This helps estimate the expected outcomes quickly! 🚀