Answer to Question #137967 in Statistics and Probability for Nish

Question

Suppose that the value of US dollar rising against the Japanese yen in a given week has a 50-50 chance. Assume that the weekly movement of the two currencies is independent over the next week. What is the probability that the value of the US dollar will rise against the Japanese yen in the majority of weeks in the next seven week period?

Answer 1

Let $X=$ the number of weeks in which the value of US dollar rising against the Japanese yen: $X\sim Bin(n, p)$

$$P(X=x)=\dbinom{n}{x}p^x(1-p)^{n-x}$$

Given $n=7, p=0.5$

$$P(X\geq4)=P(X=4)+P(X=5)+P(X=6)+$$

$$+P(X=7)=\dbinom{7}{4}0.5^4(1-0.5)^{7-4}+\dbinom{7}{5}0.5^5(1-0.5)^{7-5}+$$

$$+\dbinom{7}{6}0.5^6(1-0.5)^{7-6}+\dbinom{7}{7}0.5^7(1-0.5)^{7-7}=$$

$$=\dfrac{1}{128}(35+21+7+1)=0.5$$