A box contains 5 red balls and 3 blue balls. If two balls are drawn at random, what is the probability that both are red?

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Q: A box contains 5 red balls and 3 blue balls. If two balls are drawn at random, what is the probability that both are red?

Solution: Total ways to choose 2 balls from 8 = 8C2 = 28. Ways to choose 2 red balls = 5C2 = 10. Probability = 10/28 = 5/14.

Steps: 8

Step 1: Count the total number of balls in the box. There are 5 red balls and 3 blue balls, so total balls = 5 + 3 = 8.
Step 2: Determine how many ways we can choose 2 balls from the total of 8 balls. This is calculated using the combination formula, which is written as 8C2. The formula for combinations is n! / (r!(n-r)!), where n is the total number of items, r is the number of items to choose, and '!' denotes factorial.
Step 3: Calculate 8C2. This means we need to calculate 8! / (2!(8-2)!) = 8! / (2! * 6!) = (8 * 7) / (2 * 1) = 28. So, there are 28 ways to choose 2 balls from 8.
Step 4: Now, we need to find out how many ways we can choose 2 red balls from the 5 red balls. This is calculated using the combination formula, written as 5C2.
Step 5: Calculate 5C2. This means we need to calculate 5! / (2!(5-2)!) = 5! / (2! * 3!) = (5 * 4) / (2 * 1) = 10. So, there are 10 ways to choose 2 red balls from 5.
Step 6: Now, we can find the probability that both balls drawn are red. The probability is calculated as the number of ways to choose 2 red balls divided by the total number of ways to choose 2 balls. This is 10 / 28.
Step 7: Simplify the fraction 10/28. Both numbers can be divided by 2, so 10/28 = 5/14.
Step 8: The final answer is that the probability of drawing 2 red balls is 5/14.