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Answer :
The probability that all three selected adults are between 180 and 185 cm tall is 0.0015 or 0.15%. The probability that none of the selected adults are between 180 and 185 cm tall is 0.6611 or 66.11%.
Given that the probability of a single adult being between 180 and 185 cm is 0.1157, we can assume that this probability is the same for all adults in the population.
(a) The probability can be calculated as follows:
P(all three between 180 and 185 cm) = P(first adult between 180 and 185 cm) x P(second adult between 180 and 185 cm) x P(third adult between 180 and 185 cm)
Since the selection of each adult is independent, we can multiply the probabilities. Thus,
P(all three between 180 and 185 cm) = 0.1157 x 0.1157 x 0.1157
P(all three between 180 and 185 cm) = 0.0015 (rounded to four decimal places) or 0.15%
(b) The probability can be calculated as follows:
P(none between 180 and 185 cm) = P(first adult not between 180 and 185 cm) x P(second adult not between 180 and 185 cm) x P(third adult not between 180 and 185 cm)
Since the selection of each adult is independent, we can multiply the probabilities. Thus,
P(none between 180 and 185 cm) = (1 - 0.1157) x (1 - 0.1157) x (1 - 0.1157)
P(none between 180 and 185 cm) = 0.6611 (rounded to four decimal places) or 66.11%
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(a) The probability that all three adults are between 180 and 185 cm tall is approximately 0.00153.
(b) The probability that none of the adults are between 180 and 185 cm tall is approximately 0.739.
To solve part (a), we are looking for the probability that all three independently selected adults are between 180 and 185 cm tall. Since the probability of one adult being in this height range is 0.1157, and the events are independent, we can multiply the probabilities together to find the probability that all three are in this range:
[tex]\[ P(\text{all three}) = P(\text{first adult}) \times P(\text{second adult}) \times P(\text{third adult}) \] \[ P(\text{all three}) = 0.1157 \times 0.1157 \times 0.1157 \] \[ P(\text{all three}) \approx 0.00153 \][/tex]
For part (b), we want to find the probability that none of the adults are between 180 and 185 cm tall. This is the complement of the probability that at least one adult is in this height range. The probability that a single adult is not in this height range is 1 - 0.1157. Since the events are independent, we raise this probability to the power of 3 for the three adults:
[tex]\[ P(\text{none}) = (1 - P(\text{single adult}))^3 \] \[ P(\text{none}) = (1 - 0.1157)^3 \] \[ P(\text{none}) \approx 0.739 \][/tex]
Therefore, the probability that all three adults are between 180 and 185 cm tall is approximately 0.00153, and the probability that none of the adults are between 180 and 185 cm tall is approximately 0.739.
