Final answer:
The probability that no one in the family of six will contract the flu is 0.000064. The probability that all six family members will contract the flu is 0.262144. The probability that at least two family members will contract the flu is 0.83.
Step-by-step explanation:
To find the probability that no one will contract the flu, we need to find the probability that each individual family member will not contract the flu. Since the probability that each individual will not contract the flu is 1 - 0.8 = 0.2, the probability that no one will contract the flu is 0.2^6 = 0.000064.
To find the probability that all six family members will contract the flu, we need to find the probability that each individual will contract the flu. Since the probability that each individual will contract the flu is 0.8, the probability that all will contract the flu is 0.8^6 = 0.262144.
To find the probability that at least two family members will contract the flu, we need to find the probability that two, three, four, five, or six family members will contract the flu. We can calculate this by adding the probabilities of each of these cases. The probability that two family members will contract the flu is (0.8^2)*(0.2^4)*6!/2!(6-2)! = 0.2304. The probability that three family members will contract the flu is (0.8^3)*(0.2^3)*6!/3!(6-3)! = 0.3072. The probability that four family members will contract the flu is (0.8^4)*(0.2^2)*6!/4!(6-4)! = 0.2048. The probability that five family members will contract the flu is (0.8^5)*(0.2^1)*6!/5!(6-5)! = 0.0768. The probability that six family members will contract the flu is (0.8^6)*(0.2^0)*6!/6!(6-6)! = 0.0114. Adding up these probabilities, we get a total probability of 0.2304 + 0.3072 + 0.2048 + 0.0768 + 0.0114 = 0.83. Therefore, the probability that at least two family members will contract the flu is 0.83.