Answer:
Probability says that divide the required possible outcomes by the total number of outcome.
As per the statement:
Your friend brings 15 chocolate cupcakes and 15 vanilla cupcakes to school.
We have to find the probability that the first student will pick 2 chocolate cupcakes.
Number of required possible outcomes = \(^{15}C_{2} \times ^{15}C_{0}\)
Total number of outcomes = \(^{30}C_{2}\)
by definition we have;
\(\text{P(2 chocolate cupcakes)} = \frac{^{15}C_{2} \times ^{15}C_{0}}{^{30}C_{2}}\)
We know that:
\(^{15}C_{0} = 1\)
⇒\(\text{P(2 chocolate cupcakes)} = \frac{^{15}C_{2}}{^{30}C_{2}}\)
Using the formula:
\(^{n}C_{r} = \frac{n!}{r!(n-r)!}\)
⇒\(\text{P(2 chocolate cupcakes)} = \frac{\frac{15!}{2! \cdot 13!}}{\frac{30!}{2! \cdot 28!}}\)
⇒\(\text{P(2 chocolate cupcakes)} = \frac{\frac{15 \cdot 14 \cdot 13!}{2! \cdot 13!}}{\frac{30 \cdot 29 \cdot 28!}{2! \cdot 28!}}\)
Simplify:
\(\text{P(2 chocolate cupcakes)} = \frac{15 \cdot 14}{30 \cdot 29} = \frac{7}{29}\)
therefore, the probability that the first student will pick 2 chocolate cupcakes is, \(\frac{7}{29}\)
