NEM1002: Statistics for Decision Making - Binomial Probability

These flashcards cover key concepts, definitions, and examples related to binomial probability and its applications, as discussed in the lecture notes.

What is the binomial distribution an example of?

A discrete probability distribution.

What conditions must be satisfied for a trial to be a Bernoulli trial?

It must have n independent trials, only two possible outcomes (success and failure), and a fixed probability of success (p).

How is a binomial random variable expressed mathematically?

X ∼ Bi(n, p) where n is the number of trials and p is the probability of success.

If a fair die is rolled 4 times, how do we denote the number of times a 5 appears?

X ∼ Bi(4, 1/6).

What formula is used to calculate the probability of a binomial random variable?

Pr(X = x) = n C x * p^x * q^(n−x) where q = 1 - p.

In the example of rolling a die 4 times, what is Pr(X = 0)?

625/1296.

What is the expected value formula for a binomial distribution?

E(X) = np.

What is the variance formula for a binomial distribution?

Var(X) = npq.

In Example 1 with X ∼ Bi(6, 0.4), what is the value of n?

n = 6.

What probability distribution table values did you find for x in Example 1?

Pr(X = 0) = 0.046656, Pr(X = 1) = 0.186624, Pr(X = 2) = 0.31104, Pr(X = 3) = 0.27648, Pr(X = 4) = 0.13824, Pr(X = 5) = 0.036864, Pr(X = 6) = 0.004096.

What do you find for Pr(X ≥ 3) in Example 5 with p = 0.3 and n = 5?

Pr(X ≥ 3) = 0.16308.

In Example 6, if a bag contains 4 red and 3 blue marbles, what is the probability that all 7 selections are red?

(4/7)^7.

What is the standard deviation formula for a binomial distribution?

SD(X) = √npq.

Given that X ∼ Bi(40, 0.25), what's the expected value?

E(X) = 10.

How do you calculate the variance and standard deviation for a distribution with X ∼ Bi(40, 0.25)?

Variance: Var(X) = 7.5, Standard Deviation: SD(X) = 2.7386.

What is a potential outcome of rolling a fair die 90 times regarding the expected number of even numbers?

Expected value = n * p = 90 * 0.5 = 45.