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 even once out of 12 times can be calculated using the binomial distribution formula. If this probability is less than 1%, we can reject the hypothesis that each brother has an equal chance of 1/3.

Step-by-step explanation:

The probability of Romain not getting picked even once out of 12 times can be calculated using the binomial distribution formula. The formula is:

P(X=k) = (nCk) * p^k * (1-p)^(n-k)

Where:

• n is the number of trials (12 in this case)
• k is the number of successes (0 in this case)
• p is the probability of success (1/3 in this case)
• (nCk) is the number of combinations of n objects taken k at a time

Using this formula, we can calculate the probability of Romain not getting picked even once:

P(X=0) = (12C0) * (1/3)^0 * (2/3)^(12-0) = (1) * (1) * (2/3)^12 = (1) * (1) * 0.006 = 0.006

This means that there is a 0.6% chance of Romain not getting picked even once out of 12 times if the hypothesis of each brother having an equal chance of 1/3 is correct.

This probability is less than 1%, so we can reject the hypothesis and conclude that it is unlikely that each brother has an equal chance of 1/3.