Final answer:
To find the probability that a red ball drawn at random is from Box B, we calculate the individual probabilities of drawing a red ball from each box and apply Bayes' theorem. The probability that the red ball is from Box B is 9/14, making the answer (d).
Step-by-step explanation:
Probability of Drawing a Red Ball from Box B
The question asks for the probability that a red ball drawn at random is from Box B. First, we calculate the chance of drawing a red ball from each box individually, and then we apply Bayes' theorem to find the desired probability. Box A has a total of 5 balls (2 white and 3 red), while Box B has 9 balls (4 white and 5 red).
The probability of choosing Box A and then drawing a red ball from it is (1/2) * (3/5) = 3/10, and the probability of choosing Box B and then drawing a red ball from it is (1/2) * (5/9) = 5/18. To find the total probability of drawing a red ball from either box, we add these probabilities together:
Total probability of drawing a red ball = (3/10) + (5/18) = 54/90 + 25/90 = 79/90.
To find the probability that the red ball is from Box B given that a red ball was drawn, we divide the probability of drawing a red ball from Box B by the total probability of drawing a red ball:
Probability = (Probability of red from B) / (Total probability of red) = (5/18) / (79/90) = 5/18 * 90/79 = 450/1422, which simplifies to 5/14, or 9/14 when expressed in simplest form.
Therefore, the answer is (d) 9/14, indicating that the probability the red ball was drawn from Box B is 9/14.