A class consists of 55% boys and 45% girls. It is observed that 25% of the class are boys and scored an A on the test, and 35% of the class are girls and scored an A on the test.
If a student is chosen at random and is found to be a girl, the probability that the student scored an A is __?
since 45% are girls, and 35% are girls with A, then
P(A|girl) = 35/45 = 7/9
To find the probability that a randomly chosen girl scored an A, we can use Bayes' theorem.
Let's define the following events:
A = student scored an A
B = student is a girl
We are given the following probabilities:
P(A|B) = 35% (the probability that a girl scored an A)
P(B) = 45% (the probability that a randomly chosen student is a girl)
We want to find P(A|B), which is the probability that a student scored an A given that the student is a girl.
Using Bayes' theorem, we have:
P(A|B) = P(A∩B) / P(B)
Since P(A∩B) represents the probability that a student is a girl and scored an A, we can calculate it as follows:
P(A∩B) = P(A|B) * P(B) = 35% * 45% = 0.35 * 0.45 = 0.1575
Now, let's calculate P(B) - the probability that a randomly chosen student is a girl:
P(B) = 45%
Finally, we can calculate P(A|B) by dividing P(A∩B) by P(B):
P(A|B) = P(A∩B) / P(B) = 0.1575 / 0.45 ≈ 0.35
Therefore, the probability that a randomly chosen girl scored an A is approximately 0.35, or 35%.
To find the probability that the randomly chosen student is a girl and scored an A, we can use Bayes' theorem.
Let's denote:
A = Event that the student scored an A
G = Event that the student is a girl
We are given:
P(G) = 45% = 0.45 (probability of the student being a girl)
P(A|G) = 35% = 0.35 (probability of scoring an A given that the student is a girl)
We need to find:
P(A|G) = ? (probability of scoring an A given that the student is a girl)
We can use Bayes' theorem formula:
P(A|G) = P(G|A) * P(A) / P(G)
Now let's calculate the values needed to plug into the formula:
1. P(G|A) = Probability of being a girl given that the student scored an A.
We can find this using the given information:
P(G|A) = (P(G) * P(A|G)) / P(A)
= (0.45 * 0.35) / P(A)
2. P(A) = Probability of scoring an A.
To calculate this, we need to consider both boys and girls who scored an A.
We are given that 25% of the class are boys and scored an A, and 35% of the class are girls and scored an A.
So, the total probability of scoring an A is:
P(A) = P(A|B) * P(B) + P(A|G) * P(G)
= 0.25 * 0.55 + 0.35 * 0.45
Now, we can substitute these values into the formula to find P(A|G):
P(A|G) = (P(G|A) * P(A)) / P(G)
Replace P(G|A) = (0.45 * 0.35) / P(A) and P(A) = 0.25 * 0.55 + 0.35 * 0.45 to get the final calculation.
P(A|G) = (0.45 * 0.35) / (0.25 * 0.55 + 0.35 * 0.45)
Calculate the value of P(A|G) to find the probability that a randomly chosen girl scored an A.