Question

learning goal
i can decide if a model is consistent with results, given a data - generating process such as simulation.
lesson 23.1 checkpoint
□ once you have completed the above problems and checked your solutions, complete the lesson checkpoint below.
□ complete the lesson reflection above by circling your current understanding of the learning goal.
you flip a coin 4 times in a row. the histogram shows the theoretical probability distribution for this situation. use the histogram to answer the following questions. write all answers in simplified fraction form.

1. the probability of getting 3 or more heads is _.
2. the probability of getting at most 2 heads is _.
3. is the coin fair? explain how you know?

Explanation:

Step1: Find probability for 3 or more heads

From the histogram, the probability of 3 heads is $\frac{4}{16}$ and of 4 heads is $\frac{1}{16}$.
$$P(3\text{ or more})=\frac{4}{16}+\frac{1}{16}$$

Step2: Calculate the sum

$$\frac{4 + 1}{16}=\frac{5}{16}$$

Step3: Find probability for at most 2 heads

At most 2 heads means 0, 1, or 2 heads. Probability of 0 heads is $\frac{1}{16}$, 1 head is $\frac{4}{16}$, and 2 heads is $\frac{6}{16}$.
$$P(\text{at most }2)=\frac{1}{16}+\frac{4}{16}+\frac{6}{16}$$

Step4: Calculate the sum

$$\frac{1+4 + 6}{16}=\frac{11}{16}$$

Step5: Check if coin is fair

For a fair coin, the probability distribution of \(n = 4\) coin - flips (binomial distribution \(P(X=k)={n\choose k}p^{k}(1 - p)^{n - k}\), here \(p=\frac{1}{2}\), \(n = 4\)) gives \(P(X = k)=\frac{1}{16}{4\choose k}\).
\(P(X = 0)=\frac{1}{16}\), \(P(X=1)=\frac{4}{16}\), \(P(X = 2)=\frac{6}{16}\), \(P(X=3)=\frac{4}{16}\), \(P(X = 4)=\frac{1}{16}\), which matches the histogram.

Answer:

1. \(\frac{5}{16}\)
2. \(\frac{11}{16}\)
3. Yes, because the histogram matches the binomial probability distribution for \(n = 4\) and \(p=\frac{1}{2}\) (a fair - coin flip).