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In a family with 7 children what is the probability of having 4 boys and then 3 girls, in that order? (Exclude multiple births and assume all outcomes are equally likely).
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In a family with 7 children, what is the probability of having 4 boys and 3 girls, in any order? (Exclude multiple births and assume all outcomes are equally likely).
To find the probability of having 4 boys and 3 girls in a specific order in a family with 7 children, we can use the concept of permutations.
The total number of possible outcomes is given by the number of ways we can arrange 7 children, which is 7!.
To find the number of favorable outcomes where there are 4 boys followed by 3 girls in that specific order, we need to consider that there are 4! ways to arrange the 4 boys among themselves, and another 3! ways to arrange the 3 girls among themselves.
So, the number of favorable outcomes is 4! * 3!.
Therefore, the probability can be calculated as the number of favorable outcomes divided by the total number of possible outcomes:
Probability = (4! * 3!) / 7!
Now, let's calculate the probability step by step:
1. Calculate the factorial of 4, denoted as 4!:
4! = 4 * 3 * 2 * 1 = 24
2. Calculate the factorial of 3, denoted as 3!:
3! = 3 * 2 * 1 = 6
3. Calculate the factorial of 7, denoted as 7!:
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040
4. Substitute the values into the probability formula:
Probability = (24 * 6) / 5040
5. Simplify the calculation:
Probability = 144 / 5040
6. Convert the probability to decimal form:
Probability = 0.02857 (approximately)
Therefore, the probability of having 4 boys and then 3 girls in that specific order in a family with 7 children is approximately 0.02857.
Note: Since the probability is a decimal, it can also be expressed as the fraction 144/5040.