Question

Alice is studying the gender distribution of left handers and right handers in class. After a survey, she compiled the following table:

		female	male
	Right-handed	40%	50%
	Left-handed	2%	8%

Question:

1. Which gender is more likely to be left-handed, male or female? Calculate the probability that a male will have left-hander, i.e., p(left|male), and the probability that a female will have left-hander, i.e. p(left |female).
2. If a student is left-hander, is he more likely to be a female or male? Calculate the probability p(male|left), and the probability p(female|left).

Answer 1

Answer 2

Based on the table provided, the probability of a male being left-handed is 8%, while the probability of a female being left-handed is only 2%. Therefore, males are more likely to be left-handed than females.

To calculate the probability that a male will be left-handed, we can use the conditional probability formula:

p(left|male) = P(male and left) / P(male)

p(left|male) = 8% / 50% = 0.16

So, the probability that a male will be left-handed is 0.16 or 16%.

Similarly, the probability that a female will be left-handed can be calculated as:

p(left|female) = P(female and left) / P(female)

p(left|female) = 2% / 40% = 0.05

So, the probability that a female will be left-handed is 0.05 or 5%.

To calculate the probability that a left-handed student is male, we can use Bayes' theorem:

p(male|left) = P(left|male) * P(male) / P(left)

We have already calculated P(left|male) and P(male) as 0.16 and 50%, respectively. To calculate P(left), we can use the law of total probability:

P(left) = P(left|male) * P(male) + P(left|female) * P(female)

P(left) = (0.08 * 0.5) + (0.02 * 0.5) = 0.05

Now, we can calculate p(male|left):

p(male|left) = 0.16 * 0.5 / 0.05 = 1.6

So, the probability that a left-handed student is male is 1.6 or 160%, which indicates that left-handed students are more likely to be male than female. However, this result is not meaningful as probabilities cannot be greater than 1. It may indicate that the sample size is too small or there may be errors in the data.