A gender-selection technique is designed to increase the likelihood that a baby will be a girl. In the results of the gender-selection technique, 913 births consisted of 476 baby girls and 437 baby boys. In analyzing these results, assume that boys and girls are equally likely.

a. Find the probability of getting exactly 476 girls in 913 births.

b. Find the probability of getting 476 or more girls in 913 births. If boys and girls are equally likely, is 476 girls in 913 births unusually high?

c. Which probability is relevant for trying to determine whether the technique is effective: the result from part (a) or the result from part (b)?

d. Based on the results, does it appear that the gender-selection technique is effective?

The probability of getting exactly 476 girls in 913 births is
(Round to four decimal places as needed.)

The probability of getting exactly 476 girls in 913 births is around 0.0261. The probability of getting 476 or more girls is approximately 0.9621, indicating that the occurrence of 476 girls is not unusually high. Therefore, the gender-selection technique does not seem effective.

Part a: Probability of Exactly 476 Girls in 913 Births

We use the binomial probability formula:
[tex]P(X = k) = C(n, k) * p^k * (1-p)^{(n-k)}[/tex]
Here, n = 913, k = 476, and p = 0.5.

Using a binomial calculator, we find:
P(X = 476) ≈ 0.0261 (rounded to four decimal places)

Part b: Probability of Getting 476 or More Girls

To find this, we sum the probabilities from 476 girls to 913 girls:
P(X ≥ 476) ≈ 1 - P(X < 476)

Using a cumulative binomial distribution calculator, we get:
P(X ≥ 476) ≈ 0.9621

If boys and girls are equally likely, 476 girls in 913 births is not unusually high.

Part c: Relevant Probability for Technique Effectiveness

The probability of getting 476 or more girls (Part b) is relevant for determining the effectiveness.

Part d: Effectiveness of the Gender-Selection Technique

Based on the probability of getting 476 or more girls being approximately 0.9621, the gender-selection technique does not appear to produce a significant deviation from the expected 50% girl births. Therefore, it might not be effective.