Assume you have a basket full of socks, which have the following properties:
6 of the socks have blue stripes
8 of the socks have red polka dots
16 of the socks have blue polka dots
10 of the socks have red stripes
(a). If you take one sock out of the basket, what is the probability that it is red?
(b). What is the probability that a sock pulled from the basket has stripes?
(c). What is the probability that a sock pulled from the basket is either striped or blue?
(d). You select two socks from the basket, without replacement. What is the probability that the first one will have blue polka dots and the second one will have red stripes?
(e). You select two socks, without replacement. What is the probability that they will not match?
To calculate the probabilities for each question, we need to understand how many socks are in the basket and how many socks match the given properties.
(a) To calculate the probability that a randomly selected sock is red, we need to know the total number of socks and the number of red socks. In this case, there are 6 socks with red stripes and 10 socks with red stripes, so there are a total of 6 + 10 = 16 red socks in the basket. Therefore, the probability of selecting a red sock is 16/(6 + 8 + 16 + 10) = 16/40 = 0.4 or 40%.
(b) To calculate the probability that a randomly selected sock has stripes, we need to know the total number of socks and the number of striped socks. In this case, there are 6 socks with blue stripes and 10 socks with red stripes, so there are a total of 6 + 10 = 16 striped socks in the basket. Therefore, the probability of selecting a striped sock is 16/(6 + 8 + 16 + 10) = 16/40 = 0.4 or 40%.
(c) To calculate the probability that a randomly selected sock is either striped or blue, we need to know the total number of socks and the number of socks that are either striped or blue. In this case, there are 6 socks with blue stripes, 10 socks with red stripes, and 16 socks with blue polka dots. Therefore, there are a total of 6 + 10 + 16 = 32 socks that are either striped or blue. Therefore, the probability of selecting a sock that is either striped or blue is 32/(6 + 8 + 16 + 10) = 32/40 = 0.8 or 80%.
(d) To calculate the probability that the first selected sock has blue polka dots and the second selected sock has red stripes, we need to consider the probability of each event happening individually and then multiply those probabilities together. The probability of selecting a sock with blue polka dots as the first sock is 16/(6 + 8 + 16 + 10) = 16/40 = 0.4 or 40%. After removing that first sock, we have 39 socks left in the basket, and 10 of them have red stripes. Therefore, the probability of selecting a sock with red stripes as the second sock is 10/39.
To find the probability of both events happening, we multiply the probabilities together: 0.4 * 10/39 = 0.1 or 10%.
(e) To calculate the probability that two selected socks do not match, we need to consider the total number of pairs of socks that do not match and the total number of pairs of socks. In this case, there are 16 socks with blue polka dots, 10 socks with red stripes, 14 socks with different patterns (blue stripes and red polka dots), and 6 socks with no matching partner (blue stripes or red stripes).
Therefore, there are a total of 16 + 10 + 14 + 6 = 46 pairs of socks that do not match. We also need to consider the total number of pairs of socks, which is the number of socks choose 2 (denoted as C(40, 2)).
Using the formula for combinations, C(40, 2) = 40! / (2! * (40-2)!) = 40 * 39 / 2 = 780.
Therefore, the probability of selecting two socks that do not match is 46/780 = 0.0589 or 5.89%.