(1.4) Binomial

Q2.1) LeBron is a basketball player with a shooting percentage of 70%. If he attempts 20 shots, what is the probability that he makes at least 15? §

Binomial

The binomial distribution models the number of successes in a fixed number of Bernoulli trials. Its PMF is: $P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$ Where:

• $n$ is the number of trials.
• $p$ is the probability of success on each trial.
• Expected value (mean) = ( np )
• Variance = ( np(1-p) )

☞ If you repeat something $n$ times, and each time there is a probability $p$ of success, the binomial distribution tells you exactly how likely it is to see each number of successes.

Q2.2) Follow up: this experiment is an example of a sequence of which type of trials? What is an important assumption that we have made about these trials? §

A sequence of Bernoulli trials, which must be independent of each other. Another assumption we have implicitly made it that each attempt is identically distributed.

Q2.3) Follow up: Comment on the validity of this assumption. §

Q2.4) Follow up: For a binomial hypothesis test, why is the $\alpha$ of the test at $p=0.4$ equal to the $\beta$ of the test at $p=0.6$. §