Final answer:
The probability of selecting at least one state that is in the continental U.S. and went Republican in at least one of the four draws, with replacement, is calculated by subtracting the probability of four failures from 1. The probability of one failure is 21/50, so the probability of at least one success is 1 - (21/50)^4.
Step-by-step explanation:
The student is asking for the probability that at least one of four randomly selected states is in the continental U.S. and went Republican in the 2016 election. Given that 29 out of the 30 states that went Republican are in the continental U.S., the probability of randomly selecting such a state is 29/50. When we draw with replacement, the probability of not picking a continental U.S. state that went Republican in one draw is 21/50. To find the probability of at least one success in four draws, we first find the probability of zero successes (all failures) and subtract it from 1.
The probability of four failures in a row is (21/50)^4. Subtracting this from 1 gives us the probability of getting at least one successful pick.
Calculation:
- Calculate the probability of not picking a continental U.S. Republican state in one draw: 21/50.
- Raise this probability to the fourth power to represent four independent draws with replacement: (21/50)^4.
- Subtract the result from 1 to find the probability of at least one success: 1 - (21/50)^4.
The final answer will give the probability of selecting at least one state that is in the continental U.S. and went Republican in at least one of the four draws.