Multiple choice

In a city, there are three popular gyms: Gym X, Gym Y, and Gym Z. Gym X attracts 50% of the fitness enthusiasts, Gym Y attracts 30%, and Gym Z attracts the remaining 20%. It is known that 10% of the members at Gym X forget their access cards, 15% at Gym Y forget their access cards, and 5% at Gym Z forget their access cards. Now, if a randomly selected gym member is known to have forgotten their access card, what is the probability that the member is from Gym X?

  1. 0.38

  2. 0.42

  3. 0.48

  4. 0.52
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using Bayes' Theorem: P(X|Card) = P(Card|X)P(X) / [P(Card|X)P(X) + P(Card|Y)P(Y) + P(Card|Z)P(Z)]. P(X|Card) = (0.10 * 0.50) / (0.10 * 0.50 + 0.15 * 0.30 + 0.05 * 0.20) = 0.05 / (0.05 + 0.045 + 0.01) = 0.05 / 0.105 = 50/105 = 10/21 approx 0.476, which rounds to 0.48.

AI explanation

Using Bayes theorem, the probability that the member is from Gym X given they forgot their card is the probability of being from Gym X and forgetting the card divided by the total probability of forgetting the card. The probability of selecting Gym X and forgetting the card is 0.50 * 0.10 = 0.05, and the total probability of forgetting the card is (0.50 * 0.10) + (0.30 * 0.15) + (0.20 * 0.05) = 0.105. Dividing 0.05 by 0.105 gives the result, which is 0.48.

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