A consumer protection group reports that 25% of 5-lb bags of sugar of a certain brand are underweight. Three bags of sugar are selected at random. Assume that the report is correct.
a. What is the probability that all three bags are underweight?
b. What is the probability that none of the bags is underweight?
a. (1/4)^3 = 1/64
b. (3/4)^3 = 27/64
To find the probability in these cases, we can use the concept of probability and basic combinatorics.
a. To calculate the probability that all three bags are underweight, we need to find the intersection of the probabilities for each independent event. Since each bag's weight is independent of the others, we can multiply the individual probabilities.
The probability of a bag being underweight is given as 25%, or 0.25.
Therefore, the probability that all three bags are underweight is:
P(all three bags are underweight) = 0.25 * 0.25 * 0.25 = 0.015625
b. Similarly, to find the probability that none of the bags are underweight, we need to find the complement of the event where at least one bag is underweight.
The probability that a bag is not underweight is the complement of 0.25, which is 0.75.
Therefore, the probability that none of the bags are underweight is:
P(none of the bags are underweight) = 0.75 * 0.75 * 0.75 = 0.421875
Hence, the probability that all three bags are underweight is 0.015625, and the probability that none of the bags are underweight is 0.421875.