A bag contains 3 red, 5 black and 5 blue marbles. Four marbles are selected at random without replacement:
- the probability that all 4 are black?
-the probability that exactly two are blue and none are red?
2/5
First black marble = 5/13, second = 4/12, third = 3/11, fourth = 2/10.
The probability of all events occurring is found by multiplying the individual probabilities.
First blue = 5/13, second = 4/12, first non-red = 8/11, second non-red = 7/10.
The probability of all events occurring is found by multiplying the individual probabilities.
To find the probabilities, we need to first determine the total number of marbles and the number of favorable outcomes.
Step 1: Finding the total number of marbles
The total number of marbles in the bag is given by:
Total marbles = number of red marbles + number of black marbles + number of blue marbles
Total marbles = 3 + 5 + 5
Total marbles = 13
Step 2: Finding the number of favorable outcomes
a) Probability that all 4 marbles are black:
Since we are selecting 4 marbles and all of them must be black, the favorable outcome is selecting all 4 black marbles. So, the number of favorable outcomes is the number of black marbles, which is 5.
b) Probability that exactly 2 marbles are blue and none are red:
For this case, we need exactly 2 blue marbles and no red marbles. The favorable outcome consists of selecting 2 blue marbles and 2 marbles from the remaining black marbles. The number of favorable outcomes can be calculated as the product of the ways of selecting blue and black marbles.
Number of favorable outcomes = (ways to select 2 blue marbles) x (ways to select 2 black marbles)
To calculate the number of favorable outcomes, we can use the combination formula "nCr," which gives the number of ways to choose "r" objects from a set of "n" objects without replacement.
Number of favorable outcomes = C(5, 2) x C(5, 2)
Number of favorable outcomes = (5! / (2! x (5-2)!) ) x (5! / (2! x (5-2)!) )
Number of favorable outcomes = (5! / (2! x 3!) ) x (5! / (2! x 3!) )
Number of favorable outcomes = (5 x 4 x 3! / (2! x 3!) ) x (5 x 4 x 3! / (2! x 3!) )
Number of favorable outcomes = (5 x 4) x (5 x 4)
Number of favorable outcomes = 400
Step 3: Calculating the probabilities
a) Probability that all 4 marbles are black:
Probability = (Number of favorable outcomes) / (Total number of outcomes)
Probability = 5 / 13C4
Probability ≈ 0.01128
b) Probability that exactly 2 marbles are blue and none are red:
Probability = (Number of favorable outcomes) / (Total number of outcomes)
Probability = 400 / 13C4
Probability ≈ 0.30534
Therefore,
a) The probability that all 4 marbles are black is approximately 0.01128.
b) The probability that exactly 2 marbles are blue and none are red is approximately 0.30534.
To calculate the probabilities, we need to use the concept of combinations. The number of combinations is given by the formula nCr = n! / r!(n-r)!, where n is the total number of items and r is the number of items chosen at a time.
1. Probability that all 4 marbles are black:
There are a total of 13 marbles (3 red + 5 black + 5 blue). To find the probability of drawing all black marbles, we need to calculate the number of ways we can choose 4 marbles from the 5 black marbles, divided by the number of ways to choose 4 marbles from the total 13 marbles.
Number of ways to choose 4 black marbles = 5C4
Number of ways to choose 4 marbles from total = 13C4
Probability = (Number of ways to choose 4 black marbles) / (Number of ways to choose 4 marbles from total)
Probability = 5C4 / 13C4
2. Probability that exactly two marbles are blue and none are red:
There are still a total of 13 marbles, but now we need to consider 2 blue marbles and 0 red marbles. We need to calculate the number of ways to choose 2 blue marbles from the 5 blue marbles, multiplied by the number of ways to choose 2 marbles from the remaining 8 marbles (since we can't choose any red marbles), divided by the number of ways to choose 4 marbles from the total 13 marbles.
Number of ways to choose 2 blue marbles = 5C2
Number of ways to choose 2 marbles from the remaining 8 marbles = 8C2
Number of ways to choose 4 marbles from total = 13C4
Probability = (Number of ways to choose 2 blue marbles) * (Number of ways to choose 2 marbles from remaining) / (Number of ways to choose 4 marbles from total)
Probability = (5C2 * 8C2) / 13C4
You can use a calculator or manually calculate the combinations, and then divide the expressions to find the probabilities.