Question

A candy dish contains five brown and three red M&Ms. A child selects three M&Ms without checking the colors. Use this information to answer the questions in Exercises \(18-21 .\) What is the probability that the M&Ms are all red?

Short answer

Answer: The probability of selecting all 3 red M&Ms is 1/56.

Step-by-step solution

Determine the total number of ways the child can pick 3 chocolates

We can use the combination formula to find the total number of ways the child can pick 3 chocolates from 8: nCr = C(n, r) = n! / (r!(n-r)!) where n is the total number of objects and r is the number of objects chosen at a time. Total number of ways to pick 3 chocolates from 8: nCr = C(8, 3) = 8! / (3!(8-3)!)

Calculate the result of the combination formula for the total number of ways

Evaluate the expression: C(8, 3) = 8! / (3!(5)!) = 40320 / (6*120) = (8*7*6)/(3*2*1) = 56 So, there are 56 different ways the child can choose 3 chocolates from the dish.

Determine the number of ways the child can pick 3 red chocolates

Now, we will use the same combination formula to find the number of ways the child can choose 3 red chocolates: nCr = C(3, 3) = 3! / (3!(3-3)!)

Calculate the result of the combination formula for the number of ways to get 3 red chocolates

Evaluate the expression: C(3, 3) = 3! / (3!(0)!) = 6 / (6*1) = 1 So, there is 1 way in which the child can choose 3 red chocolates.

Calculate the probability of selecting 3 red chocolates

Finally, we will divide the number of ways to pick 3 red chocolates by the total number of ways to pick any 3 chocolates to find the probability: Probability = (Number of ways to pick 3 red chocolates) / (Total number of ways to pick 3 chocolates) Probability = 1 / 56 Therefore, the probability of the child selecting all 3 red M&Ms is 1/56.

Key concepts

Combinations in Probability

When we're tackling a probability problem involving selections or arrangements—like picking M&M's out of a dish—we often turn to combinations. Combinations are a key concept in probability, enabling us to determine the number of different ways we can make selections when the order doesn't matter.

Let's digest this with an example: imagine we have a group of letters, say A, B, and C, and we want to know how many ways we can select pairs. We could have AB, AC, or BC. Notice that AB is the same as BA since order isn't important. So, we have three combinations here.

In a mathematical sense, combinations are calculated using a formula that involves factorial calculations (more on that soon!). For our M&M's scenario, the formula is expressed as: \[ C(n, r) = \frac{n!}{r!(n-r)!} \] where \( n \) is the total number of items to choose from, and \( r \) is the number of items to select. This formula spares us from having to list out all possible selections manually, which can be quite a hassle, especially with larger numbers.

Factorial Calculation

Now, you might be wondering what '!' means in the combinations formula. This is called a factorial, and it's a fundamental building block in many probability calculations.

In essence, the factorial of a non-negative integer \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). Here's how it breaks down: \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Easy, right?

But why do we use factorials in combinations? Well, factorials help us account for all the different arrangements of items. When calculating combinations, we use factorials to consider all possible orderings but then adjust by eliminating the repetitive ones (since, as we said earlier, order doesn't matter).

This adjustment is a crucial step and is done by dividing the total arrangements (factorials) by the number of arrangements of the selected items (\( r! \)) and the number of arrangements of the non-selected items (\( (n-r)! \)), effectively removing duplicates and giving us the neat, non-repetitive count of combinations.

Probability Calculation

Lastly, after understanding combinations and factorial calculations, we can dive into the probability calculation. Probability is a way to measure how likely it is that an event will occur, compared to all possible outcomes.

In the M&M example, to find the probability of selecting all red candies, we divide the number of ways to get our desired outcome (in this case, the single way to pick all three red M&M's) by the total number of possible outcomes (the 56 ways to pick any three M&M's from the dish). The probability formula looks like this: \[ Probability = \frac{\text{Desired Outcomes}}{\text{Total Possible Outcomes}} \]

This simple ratio tells us the likelihood of our event—in other words, it quantifies our chances. The closer the probability is to 1, the more likely it is to occur; and the closer to 0, the less likely. For our sugar-craving child, the reality is somewhat disappointing: only 1 chance in 56 of snagging all red M&M's. That's what makes probability such a valuable tool—it gives us a clear-cut way to gauge the chances of different events, guiding expectations in games of chance, real-life scenarios, and yes, even candy selections.