When a person needs a minicab, it is hired from one of the three firms: X, Y and Z. Of the hirings, 40% are from X, 50% are from y and 10% are from Z. For cabs hired from X, 9% arrive late. The corresponding percentages for cab hired from firms Y and Z being 6% and 20% respectively.
Given that a call is made for a minicab and that it arrives late, find the probability that it came from Y
(6 * 50) / [(9 * 40) + (6 * 50) + (10 * 20)]
Well, let's calculate it together using some humor!
First, let's assume the total number of minicab hirings is 100 (to make the calculations easier). So, out of these 100 hirings, 40 are from firm X, 50 are from firm Y, and 10 are from firm Z.
Now, let's find out how many minicabs arrive late.
Out of the 40 hirings from firm X, 9% arrive late, which means 40 * 0.09 = 3.6 minicabs from firm X arrive late.
Out of the 50 hirings from firm Y, 6% arrive late, which means 50 * 0.06 = 3 minicabs from firm Y arrive late.
Out of the 10 hirings from firm Z, 20% arrive late, which means 10 * 0.2 = 2 minicabs from firm Z arrive late.
So, in total, there are 3.6 + 3 + 2 = 8.6 minicabs that arrive late out of the 100 hirings.
Now, let's find the probability that a minicab, which arrived late, came from firm Y.
Out of the 8.6 minicabs that arrive late, 3 are from firm Y.
So, the probability that a minicab, which arrived late, came from firm Y is 3 / 8.6 ≈ 0.3488.
Therefore, the probability that a minicab, which arrived late, came from firm Y is approximately 0.3488 or 34.88%.
Now, I hope this probability didn't keep you waiting too long, just like those minicabs from firm Y!
To find the probability that the minicab came from firm Y given that it arrived late, we can use Bayes' theorem.
Let's define the following events:
A = Minicab arrived late
B = Minicab came from firm Y
We are given the following probabilities:
P(A) = Probability that a minicab arrives late = 0.09 (9%)
P(B) = Probability that a minicab comes from firm Y = 0.5 (50%)
P(A|B) = Probability that a minicab arrives late given that it came from firm Y = 0.06 (6%)
Bayes' theorem states:
P(B|A) = (P(A|B) * P(B)) / P(A)
By substituting the given values:
P(B|A) = (0.06 * 0.5) / 0.09
Calculating this expression:
P(B|A) = 0.03 / 0.09
Simplifying:
P(B|A) = 1/3 or 0.3333 (approximately)
Therefore, the probability that the minicab came from firm Y given that it arrived late is approximately 0.3333 or 33.33%.
To find the probability that the cab came from firm Y given that it arrived late, we can use Bayes' theorem. Bayes' theorem states that:
P(Y|L) = (P(L|Y) * P(Y)) / P(L)
where:
P(Y|L) is the probability that the cab came from firm Y given that it arrived late,
P(L|Y) is the probability that the cab arrived late given that it came from firm Y,
P(Y) is the probability that a cab is hired from firm Y,
P(L) is the probability that a cab arrives late.
We already have the values for P(L|Y) (6% or 0.06) and P(Y) (50% or 0.5). Now, let's calculate P(L).
To calculate P(L), we need to consider all possible cases in which a cab arrives late, regardless of the firm it came from.
P(L) = P(L|X) * P(X) + P(L|Y) * P(Y) + P(L|Z) * P(Z)
Given that P(L|X) = 9% or 0.09, P(L|Z) = 20% or 0.2, P(X) = 40% or 0.4, and P(Z) = 10% or 0.1, we can calculate P(L).
P(L) = 0.09 * 0.4 + 0.06 * 0.5 + 0.2 * 0.1
P(L) = 0.036 + 0.03 + 0.02
P(L) = 0.086
Now, we can plug these values back into Bayes' theorem to find P(Y|L).
P(Y|L) = (0.06 * 0.5) / 0.086
Calculating this expression, we get:
P(Y|L) ≈ 0.35
Therefore, the probability that the cab came from firm Y given that it arrived late is approximately 0.35 or 35%.