Question

paul flipped a fair coin 10 times.
a. how many of the coin flips would you expect to land on \tails\? is this result guaranteed? explain.
b. paul recorded an \h\ if the coin landed on heads and a \t\ if the coin landed on tails. his results are shown.
ththttttttth
what proportion of pauls flips resulted in a tail? is this result surprising? explain.
c. how do you think the proportion of tails would change if paul flipped the coin 200 times? explain.

Explanation:

Step1: Calculate expected number of tails for part a

For a fair coin, probability of tails \(P(T)=\frac{1}{2}\). Number of flips \(n = 10\). Expected value \(E(X)=n\times P(T)\). So \(E(X)=10\times\frac{1}{2}=5\). This is not guaranteed as coin - flips are independent random events.

Step2: Calculate proportion of tails for part b

Count the number of 'T's in the sequence "THTHTTTTTTH". There are 8 'T's. Total flips \(n = 12\). Proportion \(=\frac{8}{12}=\frac{2}{3}\approx0.67\). Since the expected proportion is \(0.5\), for a small number of trials (12), random variation can cause such a result. It is not extremely surprising as sample size is small.

Step3: Explain for part c

As the number of flips \(n\) increases (from 10 to 200), by the law of large numbers, the sample proportion of tails will get closer to the theoretical probability \(P(T) = 0.5\).

Answer:

a. We would expect \(5\) tails. This result is not guaranteed because each coin - flip is an independent random event.
b. The proportion of tails is \(\frac{2}{3}\). This result is not very surprising because the sample size (\(n = 12\)) is small, and there can be significant random variation in a small number of trials.
c. The proportion of tails would get closer to \(0.5\). By the law of large numbers, as the number of trials (\(n\)) increases, the sample proportion of an event gets closer to the theoretical probability of that event.