High School

Please answer the following questions:

**QUESTION 2**

Semen from a promising bull is going to be used for artificial insemination. However, the owner is concerned that the bull may carry a recessive lethal condition. To test the bull, the owner inseminates a total of 20 cows. Cows in the herd are 85% homozygous dominant and 15% are heterozygous for the lethal gene. All cows produced healthy normal calves. What is the owner’s level of confidence that the bull is not a carrier of the lethal gene? (Use 4 decimals in your answer)

**QUESTION 3**

Semen from a promising bull is going to be used for artificial insemination. However, the owner is concerned that the bull may carry a recessive lethal condition. Cows in the herd are 55% homozygous dominant and 45% are heterozygous for the lethal gene. Using this information, calculate the number of cows that the owner has to inseminate in order to be 90% confident that the bull is not a carrier. (Assume that all cows inseminated will have healthy normal calves).

Answer :

Final answer:

The owner's level of confidence that the bull is not a carrier of the lethal gene is 97.38%. The owner needs to inseminate approximately 6 cows to be 90% confident that the bull is not a carrier.

Explanation:

In order to determine the owner's level of confidence that the bull is not a carrier of the lethal gene, we need to look at the observed results and compare them to the expected results. In this case, all 20 cows produced healthy normal calves, which is consistent with the 85% homozygous dominant and 15% heterozygous distribution in the herd. This indicates that the bull is very unlikely to be a carrier of the lethal gene. To calculate the owner's level of confidence, we can use the binomial probability formula. Given that all calves were normal, the probability of this happening by chance alone (assuming the bull is a carrier) is 0.85^20 = 0.0262. Therefore, the owner's level of confidence that the bull is not a carrier is 1 - 0.0262 = 0.9738, or 97.38%.

In question 3, to calculate the number of cows the owner needs to inseminate in order to be 90% confident that the bull is not a carrier, we can again use the binomial probability formula. We want to find the number of cows (n) that satisfies the equation 1 - (0.85^n + 0.45^n) = 0.9. Using trial and error or an iterative approach, we can find that n ≈ 6. Therefore, the owner needs to inseminate approximately 6 cows to be 90% confident that the bull is not a carrier.

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