A jar contains 12 marbles, 5 of which are blue and 7 of which are red. If 4 marbles are
chosen at random and without replacement, what is the probability of selecting 3 blue ones and 1 red
one?
A delivery truck must make stops in eight different cities, designated by the first letter in
the name of the city: A, B, C, D, E, F, G, and H. If the order in which the truck visits the eight locations
is chosen randomly, what is the probability that it will visit them in alphabetical order?
First blue = 5/12
Second blue = 4/11
Third blue = 3/10
Red = 7/9
The probability that all/both events occur is found by multiplying the probabilities of the individual events.
To find the probability of selecting 3 blue marbles and 1 red marble, we need to calculate the number of favorable outcomes divided by the total number of possible outcomes.
Step 1: Calculate the total number of possible outcomes:
We are choosing 4 marbles out of 12 without replacement, so we can use the combination formula. The total number of possible outcomes is the combination of 4 marbles out of 12:
C(12, 4) = 12! / (4! * (12-4)!) = 12! / (4! * 8!) = (12 * 11 * 10 * 9) / (4 * 3 * 2 * 1) = 11 * 5 * 3 = 165.
Step 2: Calculate the number of favorable outcomes:
We want to select 3 blue marbles out of the 5 available blue marbles, and 1 red marble out of the 7 available red marbles. We can use the combination formula for each case:
Number of favorable outcomes = C(5, 3) * C(7, 1) = (5! / (3! * (5-3)!) * (7! / (1! * (7-1)!)) = (5 * 4 / 2 * 1) * (7 * 6 * 5 * 4 / (4 * 3 * 2 * 1)) = 10 * 35 = 350.
Step 3: Calculate the probability:
The probability of selecting 3 blue marbles and 1 red marble is the number of favorable outcomes divided by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes = 350 / 165 = 2.1212 (rounded to four decimal places).
Therefore, the probability of selecting 3 blue marbles and 1 red marble is approximately 0.2121.
To find the probability of selecting 3 blue marbles and 1 red marble, we need to determine the total number of possible outcomes and the number of favorable outcomes.
1. Total number of possible outcomes:
When 4 marbles are chosen without replacement, the total number of possible outcomes can be found using the combination formula (nCr). In this case, n represents the total number of marbles (12) and r represents the number of marbles chosen (4).
So, the total number of possible outcomes = 12C4 = 12! / (4!(12-4)!) = 12! / (4! * 8!) = (12 * 11 * 10 * 9) / (4 * 3 * 2 * 1) = 495.
2. Number of favorable outcomes:
To select 3 blue marbles and 1 red marble, we can choose 3 blue marbles out of the 5 available blue marbles (5C3) and 1 red marble out of the 7 available red marbles (7C1).
So, the number of favorable outcomes = 5C3 * 7C1 = (5! / (3!(5-3)!)) * (7! / (1!(7-1)!)) = (5 * 4 / (2 * 1)) * 7 = 10 * 7 = 70.
3. Probability:
The probability can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
So, the probability of selecting 3 blue marbles and 1 red marble = number of favorable outcomes / total number of possible outcomes = 70 / 495 = 14 / 99.
Therefore, the probability of selecting 3 blue marbles and 1 red marble is 14/99.