suppose a bag contains three orange marbles and two blue marbles. you are to choose a marble, return it to the bag, and then choose again and suppose you do this 50 times what is chance of getting 2 marbles of the same color.
To determine the probability of getting 2 marbles of the same color after choosing and returning a marble 50 times from the described bag, we can use the concept of probability and combinations.
First, let's break down the problem step by step:
1. Determine the probability of choosing an orange marble on any given attempt:
- There are a total of 3 orange marbles out of 5 marbles in the bag, so the probability of getting an orange marble is 3/5.
2. Determine the probability of choosing a blue marble on any given attempt:
- There are a total of 2 blue marbles out of 5 marbles in the bag, so the probability of getting a blue marble is 2/5.
Now, we need to consider the different combinations of choosing 2 marbles of the same color out of 50 attempts. We can calculate this using the concept of binomial probability.
The probability of choosing 2 marbles of the same color is the sum of two probabilities:
1. The probability of choosing 2 orange marbles.
2. The probability of choosing 2 blue marbles.
To calculate the probability of choosing 2 orange marbles, we will use the binomial probability formula:
P(x=k) = C(n, k) * (p^k) * ((1-p)^(n-k))
Where:
P(x=k) is the probability of getting k successes (in this case, 2 orange marbles),
C(n, k) is the number of combinations of n objects taken k at a time (in this case, 50 choose 2),
p is the probability of success (in this case, probability of choosing an orange marble),
n is the total number of trials (in this case, 50).
Using the same formula, we can calculate the probability of choosing 2 blue marbles.
Finally, we sum up the probabilities of choosing 2 orange marbles and 2 blue marbles to get the overall probability of getting 2 marbles of the same color.
However, calculating this probability with the given number of attempts (50) would involve a large number of calculations. It would be impractical to solve it manually.
To get the exact probability, we can use a computer program or a statistical software that can handle large calculations.