Question

A box containing 5 yellow balls, 3 orange balls, and 1 red ball. Two balls are drawn at random. Find the probability that one ball is orange and the other is red.

Answer 1

`(1)/(12)` or 0.0833

Explanation:

This problem can be solved using the formula:

P(A∩B)=P(A)×P(B∣A)

where P(A∩B) is the probability of both events happening, P(A) is the probability of the first event happening, and P(B∣A) is the probability of the second event happening given that the first event has happened.

In this case, we want to find the probability that one ball is orange and the other is red. There are two possible ways this can happen: either we draw an orange ball first and then a red ball, or we draw a red ball first and then an orange ball. We can use the formula to calculate the probability of each scenario and then add them up.

The probability of drawing an orange ball first is P(O)=3/9​, since there are 3 orange balls out of 9 total balls. The probability of drawing a red ball second given that we have drawn an orange ball first is P(R∣O)=1/8​ since there is 1 red ball out of 8 remaining balls. Therefore, the probability of drawing an orange ball first and then a red ball second is:

P(O∩R)=P(O)×P(R∣O)= 93/9 × 1/8 ​= 1/24​

Similarly, the probability of drawing a red ball first is P(R)=1/9​, since there is 1 red ball out of 9 total balls. The probability of drawing an orange ball second given that we have drawn a red ball first is P(O∣R)=3/8​, since there are 3 orange balls out of 8 remaining balls. Therefore, the probability of drawing a red ball first and then an orange ball second is:

P(R∩O)=P(R)×P(O∣R)= 1/9 × 3/8 ​= 1/24​

Finally, we can add up the probabilities of both scenarios to get the answer:

P(O,R)=P(O∩R)+P(R∩O)= 1/24 + 1/24​ = 2/24 ​= 1/12

Therefore, the probability that one ball is orange and the other is red is 1/12​ or about 0.0833.

Answer 2

Ok, sure.

Explanation:

Total number: 5 yellow + 3 orange + 1 red = 9 balls.

To calculate the probability, we need to determine the favorable outcomes, which is the number of ways we can choose one orange ball and one red ball.

Number of ways to choose one orange ball: 3

Number of ways to choose one red ball: 1

Since we need to choose one orange ball and one red ball, the number of favorable outcomes is the product of these two:

Number of favorable outcomes = 3 * 1 = 3

Now, let's calculate the total number of possible outcomes. We will choose 2 balls out of the 9 available in the box.

Number of ways to choose 2 balls out of 9 = 9C2 = (9 * 8) / (2 * 1) = 36

Therefore, the total number of possible outcomes is 36.

Finally, we can calculate the probability:

Probability = Number of favorable outcomes / Total number of possible outcomes

Probability = 3 / 36

Probability = 1 / 12

So, the probability that one ball is orange and the other is red is 1/12.