Each day Julia's math Teacher gives class a warm up qs. It's true/false 20% of the times and multiple choice 80% . Julia gets 70% on true false correct and 90% on multiple choice correct. Julia answered today's question correctly. What's the probability that it was a multiple choice ?
.90 * .80 = ?
0.72
To find the probability that it was a multiple choice question, we can use Bayes' theorem.
Let's define the events:
A = Event that the question is a multiple choice
B = Event that Julia answered the question correctly
We are given:
P(A) = 80% = 0.8 (probability of question being multiple choice)
P(B | A) = 90% = 0.9 (probability of answering correctly given that it's a multiple choice question)
We need to find:
P(A | B) (probability that it's a multiple choice question given that Julia answered correctly)
According to Bayes' theorem:
P(A | B) = (P(B | A) * P(A)) / P(B)
To calculate P(B), we need to consider both scenarios: answering correctly on true/false and multiple choice questions.
P(B) = (P(B | A) * P(A)) + (P(B | not A) * P(not A))
P(B | not A) = 70% = 0.7 (probability of answering correctly on true/false question)
P(not A) = 1 - P(A) = 1 - 0.8 = 0.2 (probability of question not being multiple choice)
Now we can calculate P(B):
P(B) = (P(B | A) * P(A)) + (P(B | not A) * P(not A))
= (0.9 * 0.8) + (0.7 * 0.2)
= 0.72 + 0.14
= 0.86
Finally, we can calculate P(A | B):
P(A | B) = (P(B | A) * P(A)) / P(B)
= (0.9 * 0.8) / 0.86
= 0.72 / 0.86
≈ 0.84
Therefore, the probability that it was a multiple choice question given that Julia answered correctly is approximately 0.84, or 84%.
To find the probability that today's question was a multiple choice given that Julia answered it correctly, we can use Bayes' Theorem. Bayes' Theorem states:
P(A|B) = (P(B|A) * P(A)) / P(B)
Where:
- P(A|B) is the probability of event A happening given that event B has occurred.
- P(B|A) is the probability of event B happening given that event A has occurred.
- P(A) is the probability of event A occurring.
- P(B) is the probability of event B occurring.
In this case, we want to find the probability that today's question was a multiple choice given that Julia answered it correctly. Let's assign the events as follows:
- A: Today's question is a multiple choice.
- B: Julia answered the question correctly.
We are given the following probabilities:
- P(A) = 0.80 (80% of the questions are multiple choice).
- P(B|A) = 0.90 (Julia gets 90% of the multiple choice questions correct).
We need to find P(B), the probability that Julia answered the question correctly. This can be calculated by considering the total probability of Julia answering correctly for both types of questions:
- P(B) = P(A) * P(B|A) + P(not A) * P(B|not A)
Given that only two types of questions are possible (multiple choice and true/false), if P(A) represents the probability of A happening, then P(not A) represents the probability of not A happening. In this case, P(not A) = 1 - P(A) = 1 - 0.80 = 0.20.
We are not given P(B|not A) directly, but we can calculate it by subtracting P(B|A) from 1. Since Julia gets 70% of the true/false questions correct, P(B|not A) = 1 - 0.70 = 0.30.
Now we can calculate P(B):
- P(B) = P(A) * P(B|A) + P(not A) * P(B|not A)
- P(B) = 0.80 * 0.90 + 0.20 * 0.30
- P(B) = 0.72 + 0.06
- P(B) = 0.78
Finally, we can use Bayes' Theorem to calculate the probability of today's question being a multiple choice given that Julia answered it correctly:
- P(A|B) = (P(B|A) * P(A)) / P(B)
- P(A|B) = (0.90 * 0.80) / 0.78
- P(A|B) = 0.72 / 0.78
- P(A|B) ≈ 0.923 (approximately)
Therefore, the probability that today's question was a multiple choice given that Julia answered it correctly is approximately 0.923, or 92.3%.