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 $\frac{1}{3}$ of getting picked in each draw 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.
$\square$
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:
(A) We cannot reject the hypothesis.
(B) We should reject the hypothesis.

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 $\frac{1}{3}$ of getting picked in each draw 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.
$\square$
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:
(A) We cannot reject the hypothesis.
(B) We should reject the hypothesis.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the probability of Romain not being picked in 12 draws, we calculate $(\frac{2}{3})^{12}$ since there is a $\frac{2}{3}$ chance of Romain not being picked each time. ‖ ‖ step 2 ⋮ Using the asksia-ll calculator, the result of $(\frac{2}{3})^{12}$ is approximately $0.0077073466292589393742673222427324952346544583500332115888687549...$ ‖ ‖ step 3 ⋮ To express this probability as a percentage, we multiply by 100: $0.0077073466292589393742673222427324952346544583500332115888687549... imes 100 \approx 0.77\%$. ‖ ‖ step 4 ⋮ Since the probability of Romain not being picked even once out of 12 times is approximately $0.77\%$, which is less than the significance level of $1\%$, we reject the null hypothesis. ‖ ∻1 Answer∻ ***B*** ∻Key Concept∻ ⚹Hypothesis Testing and Probability⚹ ∻Explanation∻ ⚹In hypothesis testing, if the probability of an observed outcome under the null hypothesis is less than the significance level, we reject the null hypothesis. Here, the probability of Romain not being picked is less than the 1% significance level, leading to the rejection of the null hypothesis.⚹◊What is the formula for calculating the probability of Romain not getting picked even once out of 12 times if each brother has an equal chance of $\frac{1}{3}$ of getting picked in each draw?⍭ Generate me a similar question◊

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