Question

A bag contains 3 red, 4 blue, and 6 yellow marbles. One marble is selected at a time, and once a marble is selected, it is not replaced. Find the probability of choosing two yellow marbles. P(yellow, yellow)

Answer 1

The probability of choosing two yellow marbles = 5/26

Step-by-step explanation:

Step 1 :

Given

Total number of marbles in the bag = 13 marbles

Number of yellow marbles = 6.

We choose 2 marbles from the bag without replacing and need to determine the probability that both are yellow.

Step 2 :

Probability of finding yellow marble in the first try = Number of yellow marbles / Total number of marbles = $\frac{6}{13}$

Once the yellow marble is selected we have 5 yellow marbles among 12 remaining marbles in the bag.

Probability of finding yellow marble in the second try = Number of yellow marbles / Total number of marbles = $\frac{5}{12}$

Step 3:

P(yellow in first 2 tries) = P(yellow in first try) * P(yellow in second try)

P(yellow in first 2 tries) = $\frac{6}{13} * \frac{5}{12}$ = $\frac{5}{26}$

Step 4:

Answer:

The probability of choosing two yellow marbles = 5/26

Answer 2

Using the hypergeometric distribution, it is found that there is a 0.1923 = 19.23% probability of choosing two yellow marbles, that is, P(yellow, yellow) = 0.1923.

In this problem, the marbles are chosen without replacement, hence the hypergeometric distribution is used to solve this question.

What is the hypergeometric distribution formula?

The formula is:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}$

$C_{n,x} = \frac{n!}{x!(n-x)!}$

The parameters are:

• x is the number of successes.

• N is the size of the population.

• n is the size of the sample.

• k is the total number of desired outcomes.

In this problem:

• There is a total of 3 + 4 + 6 = 13 marbles, hence $N = 13$.

• 6 are yellow, hence $k = 6$.

• 2 will be chosen, hence $n = 2$.

The probability of choosing two yellow marbles is P(X = 2), hence:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}$

$P(X = 2) = h(2,13,2,6) = \frac{C_{6,2}C_{7,0}}{C_{13,2}} = 0.1923$

0.1923 = 19.23% probability of choosing two yellow marbles, that is, P(yellow, yellow) = 0.1923.

To learn more about the hypergeometric distribution, you can take a look at https://brainly.com/question/25925490