Final answer:
The probability of winning the second prize in the 6/49 lottery is calculated using combinations. The resulting probability is 1/258.
Step-by-step explanation:
The subject of the question is Mathematics, and it pertains to the topic of probability in the context of a lottery game. In the 6/49 lottery, players choose 6 numbers from a set of 49. The second prize is usually won by matching five numbers plus the bonus number.
To find the probability of winning the second prize, we use the combination formula which is used to determine the number of ways to choose a subset of objects from a larger set where order does not matter. The formula for combinations is C(n, k) = n! / [k!(n-k)!] where n is the total number of objects, and k is the number of objects to be chosen. The notation '!' denotes a factorial, the product of all positive integers up to that number.
The calculation for the probability of choosing 5 correct numbers from the 49 possible numbers is C(49, 5), and since there are 44 remaining numbers from which the bonus number must be chosen, the calculation for the bonus number is C(44, 1). Thus, the probability of winning second prize is:
P(second prize) = C(49, 5) * C(44, 1) / C(49, 6)
Calculating the factorials and performing the combinations gives:
P(second prize) = [49! / (5! * 44!)] * [44! / (1! * 43!)] / [49! / (6! * 43!)]
= [49 * 48 * 47 * 46 * 45 / (5 * 4 * 3 * 2 * 1)] * 44 / [49 * 48 * 47 * 46 * 45 * 44 / (6 * 5 * 4 * 3 * 2 * 1)]
= 1 / (43 * 6)
Simplifying this, we get:
P(second prize) = 1 / 258
Therefore, the probability of winning the second prize in the 6/49 lottery is 1/258.