Problem 13 - 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. \(P(\text { at least } 1 \text { boy })\)

Step-by-Step Solution

Answer

The probability that at least 1 boy is selected is \( \frac{55}{56} \).

Step 1: Total Possible Outcomes

Calculate the total number of ways to choose 3 friends from 8. The number of combinations is given by \[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = 56. \] So, there are 56 possible outcomes when selecting 3 friends.

Step 2: Number of Ways to Choose Only Girls

Calculate the number of ways to select 3 friends such that all are girls. Since there are only 3 girls, this scenario implies selecting all 3 girls at once, which can be represented by \[ \binom{3}{3} = 1. \] Thus, there is only 1 way to choose 3 girls.

Step 3: Number of Ways to Have at Least 1 Boy

Subtract the number of ways to choose all girls from the total number of ways to choose 3 friends. This gives us the number of combinations where at least 1 boy is included: \[ 56 - 1 = 55. \] So, there are 55 ways to choose at least one boy.

Step 4: Calculate the Probability

Now, calculate the probability that there is at least 1 boy by dividing the number of favorable outcomes by the total number of outcomes: \[ P(\text{at least 1 boy}) = \frac{55}{56}. \]

Key Concepts

Combinations

Combinations play a key role in probability and statistics, especially when determining the number of possible ways to choose items from a larger set. A combination is a selection of items where the order doesn't matter.
For example, if Jesse wants to choose 3 friends from a group of 8, the order in which he picks them does not make any difference. Two friends named Ann and Becky being picked as Ann then Becky is the same as Becky then Ann.
This rule uses the formula for combinations, which is noted by the symbol \( \binom{n}{r} \). In this formula, \( n \) is the total number of items to choose from, and \( r \) is the number of selections being made.

• Example: \( \binom{8}{3} = \frac{8!}{3!(8-3)!} \)
• This simplifies to \( \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \)

Combinations help us count the number of ways in which a subset can be selected from a larger group, which is crucial in calculating probabilities.

At Least One

The phrase 'at least one' is often used in probability and means 'one or more'. It represents the opposite of having none of the specified items.
In the problem of selecting friends, calculating the probability of choosing "at least one boy" means that out of the chosen friends, there is at least one boy in the group.
To solve this, you often need to consider the complement, which in this case is the probability of selecting no boys, or only girls. Once you know this value, you subtract it from the total possible selections to find the number of ways that include boys.

• Example: Total ways = 56, only girls = 1
• The number of ways 'at least one boy' = total ways - only girls = 56 - 1 = 55

Understanding this concept helps simplify the problem-solving process.

Binomial Coefficient

The binomial coefficient, denoted as \( \binom{n}{r} \), is a fundamental concept in combinatorics. It represents the number of ways to pick \( r \) objects from a total of \( n \) objects without regard to order.
This mathematical expression is valuable in probability calculations because it helps quantify the number of possible outcomes.
The formula for the binomial coefficient is:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

For Jesse's selection, \( \binom{8}{3} = 56 \) shows all the possible groups of 3 friends he could choose from 8 friends. The binomial coefficient is not only used in combinations but also underpins the principles of binomial distribution in statistics.

Probability Calculation

Probability calculation is the process of determining the likelihood of a certain event occurring in a set of all possible outcomes. It's calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
In Jesse's scenario, to find the probability of selecting at least one boy, we use:

• Total outcomes = 56 (all possible groups of 3 friends)
• Favorable outcomes = 55 (groups with at least 1 boy)

The probability is given by \( P(\text{at least 1 boy}) = \frac{55}{56} \).
This result tells us that almost every random selection of 3 friends will include at least one boy, since 55 out of 56 possible combinations meet this criteria. Understanding probability calculation helps assess how likely an event is and makes decision-making more informed.