Question

In this problem, we will explore probabilities from a series of events. a. If you flip 20 coins, how many would you expect to come up “heads”, on average? Would you expect every flip of 20 coins to come up with exactly that many heads? b. If you were to flip 20 coins, what would you consider a “usual” result? An “unusual” result? c. Flip 20 coins (or one coin 20 times) and record how many come up “heads”. Repeat this experiment 9 more times. Collect the data from the entire class. d. When flipping 20 coins, what is the theoretic probability of flipping 20 heads? e. Based on the class’s experimental data, what appears to be the probability of flipping 10 heads out of 20 coins? f. The formula ( ) x n x nCx p p − 1− will compute the probability of an event with probability p occurring x times out of n, such as flipping x heads out of n coins where the probability of heads is p = ½. Use this to compute the theoretic probability of flipping 10 heads out of 20 coins. g. If you were to flip 20 coins, based on the class’s experimental data, what range of values would you consider a “usual” result? What is the combined probability of these results? What would you consider an “unusual” result? What is the combined probability of these results? h. We’ll now consider a simplification of a case from the 1960s. In the area, about 26% of the jury eligible population was black. In the court case, there were 100 men on the juror panel, of which 8 were black. Does this provide evidence of racial bias in jury selection?

Answer 1

In flipping 20 coins, you'd expect to get 10 heads on average, but actual results may vary. The theoretical probability of flipping 20 heads is (0.5)20. The jury selection discrepancy between the expected and observed number of black jurors could suggest racial bias, requiring further statistical analysis.

Step-by-step explanation:

When you flip a coin 20 times, you would expect on average to get 10 heads since the theoretical probability of getting a head in any single flip is 0.5. However, not every set of 20 flips will result in exactly 10 heads due to the variability inherent in random events. A 'usual' result for flipping 20 coins might be getting a number of heads that is close to 10, such as 9, 10, or 11 heads, while an 'unusual' result would be significantly higher or lower, such as 15 heads or 5 heads.

The theoretic probability of flipping 20 heads in 20 coin tosses is (0.5)20, since each head has a probability of 0.5 and all flips are independent. For flipping 10 heads out of 20 coins based on the class's experimental data, the probability would depend on the actual results observed. By using the formula (nCx ) px (1-p)n-x which combines the binomial coefficient (the number of ways to choose x successes in n trials) with the probability of success p raised to the power of x and the probability of failure 1-p raised to the power of n-x, one can compute the theoretical probability of flipping exactly 10 heads out of 20.

Regarding the jury selection case, if 26% of the jury-eligible population is black, we would expect 26 out of 100 jurors to be black, on average. The observation that only 8 out of 100 jurors are black could suggest a deviation from this expectation which might indicate potential racial bias in jury selection. In such cases, it's important to further analyze the data statistically to confirm whether the deviation is significant enough to indicate bias rather than being due to random chance.