Binomial probability problem solving

Question

The probability of sales representative making a sale with any one customer is $\frac{1}{3}$ . The sales representative makes eight contacts a day. To find the probability of making four sales, evaluate the term

₈ $C$₄ $( \frac{1}{3} )$⁴$(\frac{2}{4})$⁴

In the expansion $(\frac{1}{3} + \frac{2}{3})$ ⁸

I know how to evaluate ₈ $C$₄ $( \frac{1}{3} )$⁴$(\frac{2}{4})$⁴ but I don't understand what they mean by " in the expansion of $(\frac{1}{3} + \frac{2}{3})$ ⁸ "

Answer 1

The binomial theorem is that for variables $a,b$ and positive integer exponent $n$, then we have the following. $$(a+b)^n = \sum_{k=0}^n {}^nC_k a^k b^{n-k}$$

Since $1=(\tfrac 13+\tfrac 23)^8$, then the term for $k=4$ corresponds to the probability of 4 successes and 4 failures among 8 independent trials.

$$1 =\sum_{k=0}^8 \mathsf P(X=k)$$

Therefore.

$$\mathsf P(X=4) ~=~ {}^8C_4\tfrac 13^4\tfrac 23^4$$ That is all.