Statistical Probability: What is it and How to Calculate it

Question 1 

Let us assume that boys (B) and girls (G) are equally likely. 

Therefore, 

The possible results are; 

BBBB, BBBG, BBGB, BBGG, BGBB, BGBG, BGGB, BGGG, GBBB, GBBG, GBGB, GBGG, GGBB, GGBG, GGGB, GGGG. 

In total, there are sixteen outcomes. 

Possible Number of Girl Children	Possible Outcomes
0	BBBB
1	BBBG, BBGB, BGBB, & GBBB.
2	BBGG, BGBG, BGGB, GBBG, GBGB, & GGBB
3	BGGG, GBGG, GGBG, & GGGB.
4	GGGG

From the table, the following probabilities are calculated. 

The probability of 1 girl 

The probability of 2 girls  

The probability of 4 girls 

The probability the third child born is a girl 

 P (BBGB, BBGG, BGGB, GBGB, BGGG, GBGG, GGGB, GGGG) 

Question 2 

The roulette shown in figure 6.11 in the textbook shows 38 numbers, 18 of which are odd, 18 are even, and two are green (Bennett, Briggs & Triola, 2018). 

The black numbers are; 

The red numbers are; 

The two green numbers are; 

The probability of spinning a green number 

The probability of spinning a number greater than 30 

Numbers greater than 30 are 31, 32, 33, 34, 35, & 36 

The probability of spinning a red number less than 10 

Red numbers less than 10 include 2,4,6, & 8. 

The probability of spinning an even black number 

Out of the 38, 0 numbers are even and black. 

The expected total of green numbers in 57,000 spins 

To determine the expected total of green numbers in 57,000 spins, the formula shown below is used. 

The expected total of green numbers in 57,000 spins = npq 

Where; 

And, 

Question 3 

Does the polling evidence support the claim that pizza declined in popularity over the month between polls? Explain why or why not. 

To prove if the polling evidence supports the claim, we have to test the hypotheses. Since the sample is large (1,500), the sampling distribution of P 1 and P 2 can be approximated through normal distribution. 

Test statistic; 

Where; 

Using a significance level of 0.05, and the P-value calculator available at; https://www.socscistatistics.com/pvalues/normaldistribution.aspx . 

The p-value for the above z value is; 

Therefore, the null hypothesis are rejected at 0.05 significance level. The polling evidence supports the claim that pizza declined in popularity over the month between polls. 

Using statistical terminology, precisely identify the population parameter the two polls were attempting to measure. How does a parameter differ from a statistic? 

The population parameter the two polls were attempting to measure is the proportions of population in the United States that like pizza. A parameter differs from a statistic in the sense that a parameter describes an entire population while a statistic describes a sample (Surbhi, 2016). 

Based on the two polls, what would you say to someone who guessed that the population parameter the polls are trying to measure is really only 50%? 

Since the population parameter being measured is significantly above 50%, the guess made is wrong. 

Question 4 

A dice has six outcomes; 1, 2, 3, 4, 5, and 6. 

When a dice is rolled twice, the sample space is; 

From the sample space above, 

Assuming everyone bets on all 10,000 rounds, what is the expected value for a person who has number 7? (Show your calculations). 

There are 6 outcomes which have a sum of 7. 

Therefore; 

The payoff=$12 if the number comes up, and -$1 if does not. 

Assuming everyone bets on all 10,000 rounds, the expected value for a person who has number 7 is calculated as shown below; 

Assuming everyone bets on all 10,000 rounds, what is the expected value for a person who has number 2? (Show your calculations). 

Assuming everyone bets on all 10,000 rounds, the expected value for a person who has number 2 is calculated as shown below; 

References 

Bennet, J., Briggs, W., & Triola, M. (2018). Statistical reasoning for everyday life (5 th ed.). Boston, MA: Pearson Education. 

Social Science Statistics. (N.d). P value from Z score calculator. [Online]. Available at: https://www.socscistatistics.com/pvalues/normaldistribution.aspx . Accessed 25 th May 2019. 

Surbhi, S. (2016). Difference between statistic and parameter. [Online]. Available at: https://keydifferences.com/difference-between-statistic-and-parameter.html . Accessed 25 th May 2019.