A recent survey of residents in Texas concluded that 55% of Austin city residents and 46% of Houston city residents broke a bone at some point during their childhood.
a. Let’s say Austin has 5200 residents, and Houston has 4500. You independently choose two residents with replacement from the combined 9700 and learn that both of them broke a bone as a child. What’s the probability that the two residents live in different cities?
For this part, I'm calculating the probability of getting one person w/ broken bone from each city (0.253), divided by the probability that any two residents who broke their bone are chosen (0.2583). However, I realized that I may be able to format this as a Bayes theorem problem instead. Which way would be correct, and if using Bayes, how would I get the right terms?
b. You interview 500 Houston residents, sampled independently with replacement. What’s the approximate probability that more than 220 of them broke a bone as a child?
I calculated this as a Bin(500, 0.46) modeled as a Normal distribution, and calculated that P(X > 220) = 0.8152. However, this probability seems off to me.