Mega Millions In Mega Millions, an urn contains balls numbered 1–56, and a second urn contains balls numbered 1–46. From the first urn, 5 balls are chosen randomly, without replacement and without regard to order. From the second urn, 1 ball is chosen randomly. For a \$1 bet, a player chooses one set of five numbers to match the balls selected from the first urn and one number to match the ball selected from the second urn. To win, all six numbers must match; that is, the player must match the first 5 balls selected from the first urn and the single ball selected from the second urn. What is the probability of winning the Mega Millions with a single ticket?

Verified step by step guidance

1. Identify the total number of ways to choose 5 balls from the first urn containing 56 balls. Since the order does not matter and there is no replacement, use the combination formula: \(\displaystyle \binom{56}{5} = \frac{56!}{5! \times (56-5)!}\).
2. Determine the total number of ways to choose 1 ball from the second urn containing 46 balls. Since only one ball is chosen, this is simply 46 possible outcomes.
3. Calculate the total number of possible outcomes for the entire Mega Millions draw by multiplying the number of combinations from the first urn by the number of choices from the second urn: \(\displaystyle \binom{56}{5} \times 46\).
4. Recognize that there is only one winning combination that matches the player's chosen 5 balls and the single ball from the second urn exactly.
5. Compute the probability of winning by dividing the number of winning outcomes (1) by the total number of possible outcomes: \(\displaystyle \text{Probability} = \frac{1}{\binom{56}{5} \times 46}\).

Key Concepts

Combinations

Combinations refer to the selection of items from a larger set where order does not matter. In this problem, choosing 5 balls from 56 without regard to order is a combination, calculated using the formula nCr = n! / [r!(n-r)!]. This concept helps determine the total number of possible 5-ball sets from the first urn.

Probability of Independent Events

When two events occur independently, the probability of both happening is the product of their individual probabilities. Here, selecting 5 balls from the first urn and 1 ball from the second urn are independent events, so their probabilities multiply to find the overall chance of winning.

Without Replacement Sampling

Sampling without replacement means once an item is chosen, it cannot be selected again. This affects the total number of possible outcomes and is crucial for calculating combinations accurately, as the pool of available balls decreases with each selection.