Question
Find the probability of the given event. You must show all steps, and you may give your final answer as an exact value, a decimal to 3 places, or a percentage to the nearest tenth.
A gumball machine has gumballs of 4 different flavors: sour apple (A), grape (G), orange (O), and cherry (C). There are six gumballs of each flavor. When 50 cents is put into the machine, two random gumballs come out. Find the probability both gumballs are cherry.
A gumball machine has gumballs of 4 different flavors: sour apple (A), grape (G), orange (O), and cherry (C). There are six gumballs of each flavor. When 50 cents is put into the machine, two random gumballs come out. Find the probability both gumballs are cherry.
Answers
GPT-4o mini
To find the probability that both gumballs dispensed from the machine are cherry, we'll follow these steps:
1. **Determine the total number of gumballs.**
- Since there are 4 different flavors (sour apple, grape, orange, cherry) and 6 gumballs of each flavor, the total number of gumballs in the machine is:
\[
4 \text{ flavors} \times 6 \text{ gumballs per flavor} = 24 \text{ gumballs}
\]
2. **Define the event of interest.**
- We want to find the probability of drawing 2 cherry gumballs.
3. **Calculate the total number of ways to choose 2 gumballs from 24.**
- The number of ways to choose 2 gumballs from a total of 24 is given by the combination formula:
\[
\binom{n}{r} = \frac{n!}{r!(n-r)!}
\]
where \( n \) is the total number of gumballs and \( r \) is the number of gumballs to choose.
- Thus, the total number of combinations to choose 2 gumballs from 24 is:
\[
\binom{24}{2} = \frac{24!}{2!(24-2)!} = \frac{24 \times 23}{2 \times 1} = 276
\]
4. **Calculate the number of ways to choose 2 cherry gumballs from the 6 cherry gumballs.**
- The number of ways to choose 2 gumballs from the 6 cherry gumballs is:
\[
\binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15
\]
5. **Calculate the probability that both gumballs are cherry.**
- The probability \( P \) that both gumballs drawn are cherry is the ratio of the number of ways to choose 2 cherry gumballs to the total number of ways to choose 2 gumballs:
\[
P(\text{both gumballs are cherry}) = \frac{\text{Number of ways to choose 2 cherry}}{\text{Total ways to choose 2 gumballs}} = \frac{15}{276}
\]
6. **Simplify the probability.**
- We can simplify \( \frac{15}{276} \) by dividing the numerator and the denominator by their greatest common divisor, which is 3:
\[
\frac{15 \div 3}{276 \div 3} = \frac{5}{92}
\]
7. **Convert to decimal or percentage (optional).**
- To express this as a decimal:
\[
\frac{5}{92} \approx 0.054347826087
\]
- Rounded to three decimal places, this is:
\[
0.054
\]
- As a percentage, this is:
\[
0.054 \times 100 \approx 5.4\%
\]
Thus, the final answer for the probability that both gumballs are cherry is:
\[
\frac{5}{92} \quad \text{or} \quad 0.054 \quad \text{or} \quad 5.4\%
\]
1. **Determine the total number of gumballs.**
- Since there are 4 different flavors (sour apple, grape, orange, cherry) and 6 gumballs of each flavor, the total number of gumballs in the machine is:
\[
4 \text{ flavors} \times 6 \text{ gumballs per flavor} = 24 \text{ gumballs}
\]
2. **Define the event of interest.**
- We want to find the probability of drawing 2 cherry gumballs.
3. **Calculate the total number of ways to choose 2 gumballs from 24.**
- The number of ways to choose 2 gumballs from a total of 24 is given by the combination formula:
\[
\binom{n}{r} = \frac{n!}{r!(n-r)!}
\]
where \( n \) is the total number of gumballs and \( r \) is the number of gumballs to choose.
- Thus, the total number of combinations to choose 2 gumballs from 24 is:
\[
\binom{24}{2} = \frac{24!}{2!(24-2)!} = \frac{24 \times 23}{2 \times 1} = 276
\]
4. **Calculate the number of ways to choose 2 cherry gumballs from the 6 cherry gumballs.**
- The number of ways to choose 2 gumballs from the 6 cherry gumballs is:
\[
\binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15
\]
5. **Calculate the probability that both gumballs are cherry.**
- The probability \( P \) that both gumballs drawn are cherry is the ratio of the number of ways to choose 2 cherry gumballs to the total number of ways to choose 2 gumballs:
\[
P(\text{both gumballs are cherry}) = \frac{\text{Number of ways to choose 2 cherry}}{\text{Total ways to choose 2 gumballs}} = \frac{15}{276}
\]
6. **Simplify the probability.**
- We can simplify \( \frac{15}{276} \) by dividing the numerator and the denominator by their greatest common divisor, which is 3:
\[
\frac{15 \div 3}{276 \div 3} = \frac{5}{92}
\]
7. **Convert to decimal or percentage (optional).**
- To express this as a decimal:
\[
\frac{5}{92} \approx 0.054347826087
\]
- Rounded to three decimal places, this is:
\[
0.054
\]
- As a percentage, this is:
\[
0.054 \times 100 \approx 5.4\%
\]
Thus, the final answer for the probability that both gumballs are cherry is:
\[
\frac{5}{92} \quad \text{or} \quad 0.054 \quad \text{or} \quad 5.4\%
\]