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 = [tex]^{15}C_{2} \times ^{15}C_{0}[/tex]
Total number of outcomes = [tex]^{30}C_{2}[/tex]
by definition we have;
[tex]\text{P(2 chocolate cupcakes)} = \frac{^{15}C_{2} \times ^{15}C_{0}}{^{30}C_{2}}[/tex]
We know that:
[tex]^{15}C_{0} = 1[/tex]
⇒[tex]\text{P(2 chocolate cupcakes)} = \frac{^{15}C_{2}}{^{30}C_{2}}[/tex]
Using the formula:
[tex]^{n}C_{r} = \frac{n!}{r!(n-r)!}[/tex]
⇒[tex]\text{P(2 chocolate cupcakes)} = \frac{\frac{15!}{2! \cdot 13!}}{\frac{30!}{2! \cdot 28!}}[/tex]
⇒[tex]\text{P(2 chocolate cupcakes)} = \frac{\frac{15 \cdot 14 \cdot 13!}{2! \cdot 13!}}{\frac{30 \cdot 29 \cdot 28!}{2! \cdot 28!}}[/tex]
Simplify:
[tex]\text{P(2 chocolate cupcakes)} = \frac{15 \cdot 14}{30 \cdot 29} = \frac{7}{29}[/tex]
therefore, the probability that the first student will pick 2 chocolate cupcakes is, [tex]\frac{7}{29}[/tex]
