Questions: The probability that a person in the United States has type B'+ blood is 896 . Five unrelated people in the United States are selected at random. Complete parts (a) through (d) (a) Find the probability that all five have type B^+ blood. The probability that all five have type B^+ blood is 0.000003 (Round to six decimal places as needed.) (b) Find the probability that none of the five have type B'+ blood. The probability that none of the five have type B^+ blood is 0.659 (Round to three decimal places as needed.) (c) Find the probability that at least one of the five has type B^+ blood. The probability that at least one of the five has type B^+ blood is (Round to three decimal places as needed.)

Solution Steps

To find the probability that all five selected individuals have type \( B^+ \) blood, we use the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where:

• \( n = 5 \) (the number of trials),
• \( x = 5 \) (the number of successes),
• \( p = 0.0896 \) (the probability of success),
• \( q = 1 - p = 0.9104 \) (the probability of failure).

Calculating this gives:

\[ P(X = 5) = \binom{5}{5} \cdot (0.0896)^5 \cdot (0.9104)^0 = 1 \cdot 6e-06 \cdot 1 = 6e-06 \]

Thus, the probability that all five have type \( B^+ \) blood is \( 6e-06 \).

Next, we calculate the probability that none of the five individuals have type \( B^+ \) blood. Again, we use the binomial probability formula:

\[ P(X = 0) = \binom{5}{0} \cdot p^0 \cdot q^{5} = 1 \cdot 1 \cdot (0.9104)^5 \]

Calculating this gives:

\[ P(X = 0) = 1 \cdot 1 \cdot 0.625 = 0.625 \]

Thus, the probability that none of the five have type \( B^+ \) blood is \( 0.625 \).

To find the probability that at least one of the five individuals has type \( B^+ \) blood, we can use the complement rule:

\[ P(\text{at least one}) = 1 - P(\text{none}) \]

Substituting the value we found in Step 2:

\[ P(\text{at least one}) = 1 - 0.625 = 0.375 \]

Thus, the probability that at least one of the five has type \( B^+ \) blood is \( 0.375 \).

Final Answer