Problem 12 - Algebra 2

Question

Jesse has eight friends who have volunteered to help him with a school fundraise. Five are boys and 3 are girls. If he randomly selects 3 friends to help him, find each probability. \(P(\text { at least } 2 \text { girls) }\)

Step-by-Step Solution

Answer

The probability of selecting at least 2 girls is \( \frac{2}{7} \).

Step 1: Calculate Total Ways to Select 3 Friends

Calculate the total number of ways to choose 3 friends from 8. This is a combination problem, where order does not matter. Use the formula for combinations: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Here \( n = 8 \) and \( r = 3 \).\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56.\]

Step 2: Calculate Ways to Select Exactly 2 Girls

To choose exactly 2 girls, select 2 girls from the 3 available. After choosing the girls, select 1 boy from the 5 available boys. Calculate each separately using the combination formula: \( \binom{3}{2} \) for girls and \( \binom{5}{1} \) for boys.\[ \binom{3}{2} = 3, \quad \binom{5}{1} = 5.\]Thus, \( 3 \times 5 = 15 \) ways to select exactly 2 girls and 1 boy.

Step 3: Calculate Ways to Select Exactly 3 Girls

Now, calculate the number of ways to select exactly 3 girls. Since there are only 3 girls, the combination is straightforward: \( \binom{3}{3} = 1 \). This means there is 1 way to select all 3 girls.

Step 4: Add Ways to Find Total for 'At Least 2 Girls'

Add the number of ways to achieve at least 2 girls: sums from Steps 2 and 3.\[ 15\text{ (exactly 2 girls) } + 1\text{ (exactly 3 girls) } = 16.\]

Step 5: Calculate Probability of 'At Least 2 Girls'

Finally, calculate the probability by dividing the favorable outcomes (from Step 4) by the total possible outcomes (from Step 1).\[ P(\text{at least 2 girls}) = \frac{16}{56} = \frac{2}{7}.\]

Key Concepts

Combinations

Combinations are a fundamental part of probability, especially when you want to find out how many ways you can choose items from a group without considering the order. In Jesse's scenario, we want to select 3 friends from a total of 8. Mathematical combinations are represented using the formula: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]where:

• \(n\) is the total number of items to choose from,
• \(r\) is the number of items to choose,
• \(!\) indicates a factorial, which means you multiply a number by all the positive integers below it.

In this problem, \(n = 8\) and \(r = 3\). After substituting these values into the formula, we calculate:\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56. \]This tells us that there are 56 potential combinations for selecting any 3 friends from a set of 8.

Favorable Outcomes

Favorable outcomes refer to the specific arrangements that satisfy the conditions we're interested in. In Jesse's case, we want the scenarios that result in 'at least 2 girls' being selected among the 3 friends. Let's explore these favorable scenarios:

• First, how can we have exactly 2 girls? Select 2 girls from the 3 available, then choose 1 boy from the 5 boys. Using combinations:\[ \binom{3}{2} = 3 \] (for girls) and \[ \binom{5}{1} = 5 \] (for boys), we find 3 combinations of girls and 5 combinations of boys, which results in \[ 3 \times 5 = 15 \] ways to select 2 girls and 1 boy.
• Next, select exactly 3 girls, using all the available girls:\[ \binom{3}{3} = 1. \] There's only one way to choose all three girls.

Combining these situations gives us 16 favorable outcomes: 15 scenarios where there are 2 girls and 1 boy, and 1 scenario where there are 3 girls.

Total Outcomes

The total outcomes in probability refer to all the possible scenarios or combinations that can occur. In Jesse’s situation, it means all the different ways you could select any group of 3 friends from the group of 8 friends. We calculated this earlier using combinations: \[ \binom{8}{3} = 56. \]This number gives us a complete picture of the set of possibilities available when picking 3 friends at random. Every outcome has an equal chance of being selected, which is crucial for calculating probability. The total number of outcomes is the denominator in our probability fraction, allowing us to measure how likely the favorable scenarios are.

At Least Condition

The 'at least' condition in probability problems indicates a minimum threshold that must be met. In Jesse's problem, we are interested in groups that have "at least 2 girls." This means we accept scenarios with 2 girls or 3 girls.To solve such problems:- First, calculate scenarios fulfilling all lesser counts. Calculate how many combinations meet the 'at least' requirement. Here, it means considering both 2 and 3 girls.- Add these favorable outcomes:\[ 15 \text{ (exactly 2 girls) } + 1 \text{ (exactly 3 girls) } = 16. \]Finally, we determine the probability by placing these favorable outcomes over the total number of outcomes:\[ P(\text{at least 2 girls}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{16}{56} = \frac{2}{7}. \] This result tells us how likely it is to randomly select a group of 3 friends that includes at least 2 girls.