Question

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?

hints;

The "favorable outcome" is to get 4 winning numbers and 2 non-winning numbers.

How many ways can we get 4 winning numbers out of the 6 possible numbers?

How many ways can we get 2 non-winning numbers out of all of the non-winning numbers?

If we multiply those numbers together, we will get the favorable outcomes.

For total outcomes, we need to see how many ways we can pick our 6 numbers.

can you explain step by step

Answer 1

No. total outcomes =   

4 winning numbers are fixed 4 numbers. So in order to pick 6 numbers that contain those 4 fixed numbers, the free choices are only of remaining 2 numbers. Hence ..

No.of favorable outcomes = No.of ways of picking 6 numbers such that a pre-specified group of 4 numbers are always in the list = No.of ways of choosing 2 numbers from the remaining 44 numbers =

So. Winning probability = /   = 1 / 12972 = 0.000077

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Answer 2

Sure! Let's break down the steps to calculate the probability of winning the third prize in the lottery game:

Step 1: Determine the number of favorable outcomes. To win the third prize, the player needs to pick 4 winning numbers and 2 non-winning numbers. Let's calculate the number of ways to do that:

Number of ways to pick 4 winning numbers out of 6 possible winning numbers: This is a combination problem, and we can use the formula for combinations: C(n, r) = n! / (r! * (n-r)!)

In this case, n (total possible winning numbers) = 6, and r (number of winning numbers to be picked) = 4. C(6, 4) = 6! / (4! * (6-4)!) = 6! / (4! * 2!) = (6 * 5 * 4 * 3) / (4 * 3 * 2 * 1) = 15

Number of ways to pick 2 non-winning numbers out of all non-winning numbers: In this case, the total number of non-winning numbers is (48 - 6) = 42. Since the order of picking the non-winning numbers does not matter, we can again use the combination formula. C(42, 2) = 42! / (2! * (42-2)!) = 42! / (2! * 40!) = (42 * 41) / (2 * 1) = 861

Now, the number of favorable outcomes (picking 4 winning numbers and 2 non-winning numbers) is the product of the two combinations: Number of favorable outcomes = 15 * 861 = 12,915

Step 2: Determine the number of total outcomes. The player needs to pick 6 numbers out of 48 possible numbers. This is again a combination problem: Total number of ways to pick 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) = 12,271,512

Step 3: Calculate the probability of winning the third prize. Probability = Number of favorable outcomes / Number of total outcomes Probability = 12,915 / 12,271,512 ≈ 0.00105

So, the probability of winning the third prize in the lottery game is approximately 0.00105 or 0.105%.