There are three different types of circus prizes marked big (B)(𝐵) , medium (M)(𝑀) and little (L)(𝐿) . Each contains a certain number of red (R)(𝑅) and gold (G)(𝐺) balls, distributed as followsbig prize (B)(𝐵) : 7R7𝑅 and 6G6𝐺 medium prize (M)(𝑀) : 6R6𝑅 and 3G3𝐺 little prize (L)(𝐿) : 3R3𝑅 and 1G1𝐺 Your friend wins 3 big prizes, 1 medium prize and 2 little prizes. Without looking, you randomly reach into one of her prizes, and randomly take out one of its balls, which happens to be gold (G)(𝐺) . Calculate the probability that you were choosing from a big prize bag.P(B|G)=𝑃(𝐵|𝐺)=
Question
There are three different types of circus prizes marked big (B)(𝐵) , medium (M)(𝑀) and little (L)(𝐿) . Each contains a certain number of red (R)(𝑅) and gold (G)(𝐺) balls, distributed as followsbig prize (B)(𝐵) : 7R7𝑅 and 6G6𝐺 medium prize (M)(𝑀) : 6R6𝑅 and 3G3𝐺 little prize (L)(𝐿) : 3R3𝑅 and 1G1𝐺 Your friend wins 3 big prizes, 1 medium prize and 2 little prizes. Without looking, you randomly reach into one of her prizes, and randomly take out one of its balls, which happens to be gold (G)(𝐺) . Calculate the probability that you were choosing from a big prize bag.P(B|G)=𝑃(𝐵|𝐺)=
Solution
To solve this problem, we need to use Bayes' theorem, which is a way to find a probability when we know certain other probabilities. The formula is:
P(A|B) = P(B|A) * P(A) / P(B)
Where:
- P(A|B) is the probability we are trying to find: the probability of choosing from a big prize bag given that we drew a gold ball.
- P(B|A) is the probability of drawing a gold ball from a big prize bag.
- P(A) is the probability of choosing a big prize bag.
- P(B) is the total probability of drawing a gold ball.
Let's calculate each of these:
-
P(B|A): The big prize bag contains 7 red balls and 6 gold balls, so the probability of drawing a gold ball from a big prize bag is 6 / (7 + 6) = 6/13.
-
P(A): Your friend won 3 big prizes, 1 medium prize, and 2 little prizes, so the probability of choosing a big prize bag is 3 / (3 + 1 + 2) = 3/6 = 1/2.
-
P(B): This is the total probability of drawing a gold ball, which we can find by adding up the probabilities for each type of bag:
- Big bag: (6/13) * (1/2) = 3/13
- Medium bag: (3/9) * (1/6) = 1/18
- Little bag: (1/4) * (1/3) = 1/12 Adding these up, P(B) = 3/13 + 1/18 + 1/12 = 0.308
Now we can plug these into Bayes' theorem and solve:
P(A|B) = (6/13) * (1/2) / 0.308 = 0.231
So the probability that you were choosing from a big prize bag, given that you drew a gold ball, is approximately 0.231 or 23.1%.
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