In a lottery game, a player picks six numbers from 1 to 48. If 4 of those 6 numbers match those drawn, the player wins third prize. What is the probability of winning this prize? (Give your answer as a fraction.)
My Answer : 1/12972
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To find the probability of winning the third prize in the lottery game, we need to determine the number of successful outcomes (winning possibilities) and the total number of possible outcomes (all the combinations).
The number of successful outcomes is determined by choosing 4 numbers correctly out of the 6 numbers drawn. We can calculate this using the combination formula:
C(n, k) = n! / (k!(n-k)!)
In this case, n = 6 (the total number of numbers drawn), and k = 4 (the number of numbers the player needs to match). Therefore:
C(6, 4) = 6! / (4!(6-4)!) = 6! / (4! * 2!) = (6 * 5 * 4 * 3 * 2 * 1) / [(4 * 3 * 2 * 1) * (2 * 1)] = 15
So, there are 15 successful outcomes.
To calculate the total number of possible outcomes, we need to determine the number of ways to choose any 6 numbers out of 48:
C(48, 6) = 48! / (6!(48-6)!) = 48! / (6! * 42!) = (48 * 47 * 46 * 45 * 44 * 43) / [(6 * 5 * 4 * 3 * 2 * 1)]
So, there are (48 * 47 * 46 * 45 * 44 * 43) / [(6 * 5 * 4 * 3 * 2 * 1)] possible outcomes.
Finally, to calculate the probability, we divide the number of successful outcomes by the total number of possible outcomes:
Probability = Successful outcomes / Total outcomes
Probability = 15 / [(48 * 47 * 46 * 45 * 44 * 43) / (6 * 5 * 4 * 3 * 2 * 1)]
Simplifying this expression gives:
Probability = 1 / 12972
Therefore, the probability of winning the third prize is 1/12972.