Let the random variable x represent the number of girls in a family with three children. Assume the probability of a child
being a girl is 0.46. The table describes the probability of having x number of girls.Determine whether the table describes
a probability distribution. If it does, find the mean and standard deviation. Is it unusual for a family of three children to
consist of three girls
x P(x)
0 0.157
1 0.402
2 0.343
3 0.098
To determine whether the table describes a probability distribution, we need to check two conditions:
1. The sum of all the probabilities should equal 1.
2. All the probabilities should be non-negative.
Let's check these conditions for the given table:
1. Sum of probabilities: Add up all the probabilities for each value of x. In this case, 0.157+0.402+0.343+0.098 = 1. Therefore, the sum of all the probabilities is indeed 1.
2. Non-negative probabilities: Look at the probabilities listed in the table. All the probabilities (0.157, 0.402, 0.343, 0.098) are non-negative. Hence, this condition is also satisfied.
Since the table satisfies both conditions, it describes a probability distribution.
Now, let's find the mean and standard deviation for this probability distribution.
Mean (Expected Value):
The mean is calculated by multiplying each value of x by their corresponding probability and summing them up. Using the formula for the mean, we get:
Mean (µ) = Σ(x * P(x))
µ = (0 * 0.157) + (1 * 0.402) + (2 * 0.343) + (3 * 0.098)
µ ≈ 0 + 0.402 + 0.686 + 0.294
µ ≈ 1.382
Therefore, the mean (expected number of girls) is approximately 1.382.
Standard Deviation:
The standard deviation is calculated using the formula:
Standard Deviation (σ) = √[Σ((x-µ)^2 * P(x))]
To find the standard deviation, we need to calculate (x-µ)^2 * P(x) for each value of x, sum them up, and then take the square root. Using the formula, we get:
σ = √[((0-1.382)^2 * 0.157) + ((1-1.382)^2 * 0.402) + ((2-1.382)^2 * 0.343) + ((3-1.382)^2 * 0.098)]
σ ≈ √[(1.911^2 * 0.157) + (-0.382^2 * 0.402) + (0.618^2 * 0.343) + (1.618^2 * 0.098)]
σ ≈ √[0.597 + 0.058 + 0.130 + 0.248]
σ ≈ √1.033
σ ≈ 1.017
Therefore, the standard deviation is approximately 1.017.
To answer the last part of the question, whether it is unusual for a family of three children to consist of three girls, we can compare the probability with some criteria. Usually, if the probability of an event occurring is less than or equal to 0.05 (or 5%), it can be considered unusual.
In this case, the probability of having three girls is 0.098, which is less than 0.05. Thus, it is considered unusual for a family of three children to consist of three girls.