Question

Alexandre has two brothers: Hugo and Romain. Every day Romain draws a

name out of a hat to randomly select one of the three brothers to wash the
dishes. Alexandre suspected that Romain is cheating, so he kept track of
the draws, and found that out of 12 draws, Romain didn't get picked even once
Let's test the hypothesis that each brother has an equal chance of 1/3 versus the alternative that Romain's
probability is lower.
Assuming the hypothesis is correct, what is the probability of Romain not getting picked even once out of 12 times? Round your answer, if
necessary, to the nearest tenth of a percent.
Let's agree that if the observed outcome has a probability less than 1%
under the tested hypothesis, we will reject the hypothesis.
What should we conclude regarding the hypothesis?
Choose 1 answer:
B
We cannot reject the hypothesis.
We should reject the hypothesis.

Answer 1

The probability of Romain not getting picked to wash the dishes 12 times in a row, given an equal chance for all brothers, is approximately 0.79%. Since this is less than 1%, we should reject the hypothesis that the draws were fair and equally likely.

Step-by-step explanation:

Alexandre's hypothesis is that each brother has an equal chance of 1/3 to be picked to wash the dishes. To find the probability of Romain not getting picked even once out of 12 times, we assume the hypothesis is correct and each draw is independent. The probability of not picking Romain in one draw is 2/3, since either Alexandre or Hugo could be picked instead.

Using the formula for independent events, we calculate the probability of Romain not being picked across multiple draws. The probability of Romain not being picked 12 times in a row is (2/3)12.

The calculated probability is approximately (2/3)^12 = 0.0079, or 0.79%. Since this is less than 1%, we conclude that the probability of Romain not being picked even once out of 12 times by pure chance is highly unlikely, and therefore, we should reject the hypothesis that each brother had an equal chance of being selected.