(a) To calculate the death probabilities for ages 50, 51, and 52, we need to use the survival probabilities given in the table. The death probability for age n is equal to the difference between the survival probability for age n and the survival probability for age n+1. For example, the death probability for age 50 is:

Death probability at age 50 = Survival probability at age 50 – Survival probability at age 51 = 0.92940 – 0.92472 = 0.00468

Similarly, the death probabilities for ages 51 and 52 are:

Death probability at age 51 = Survival probability at age 51 – Survival probability at age 52 = 0.92472 – 0.91961 = 0.00511

Death probability at age 52 = Survival probability at age 52 – Survival probability at age 53 = 0.91961 – 0.91406 = 0.00555

Therefore, the death probabilities for ages 50, 51, and 52 are 0.00468, 0.00511, and 0.00555, respectively.

(b) To calculate the minimum premium an insurance company should charge for a $1 million three-year term life insurance contract issued to a 50-year-old man, we can use the actuarial present value of the death benefits. The actuarial present value is the present value of the expected future death benefits, discounted at the risk-free interest rate. We can assume that death always takes place halfway through a year, so we need to discount the death benefits at the semiannual interest rate.

First, we need to calculate the expected future death benefits for each year of the contract. If the man dies in the first year, the death benefit is $1 million. If he dies in the second year, the death benefit is also $1 million. If he dies in the third year, the death benefit is $1 million plus the present value of the guaranteed renewal premium for a new three-year term life insurance contract at age 53. We can assume that the guaranteed renewal premium is equal to the minimum premium that the insurance company needs to charge to break even, which is the actuarial present value of the death benefit for a three-year term life insurance contract issued to a 53-year-old man.

The actuarial present value of a $1 million death benefit for a three-year term life insurance contract issued to a 53-year-old man can be calculated as follows:

PV = $1 million/(1 + 0.04/2)^6 = $863,837.88

Therefore, the expected future death benefits for each year of the contract are:

Year 1: $1 million Year 2: $1 million Year 3: $1 million + $863,837.88 = $1,863,837.88

The actuarial present value of the expected future death benefits can be calculated as follows:

APV = $1 million/(1 + 0.04/2)^5 + $1 million/(1 + 0.04/2)^6 + $1,863,837.88/(1 + 0.04/2)^6 = $2,670,846.84

Therefore, the minimum premium that the insurance company needs to charge to break even is $2,670,846.84, which is the actuarial present value of the expected future death benefits.