1.Show the outcomes, the number of combinations, and the probabilities for the discrete random variable for the number of tails in three successive flips of a coin. Also calculate the mean and the variance.
2.Baseball’s World Series follows a best-of-seven series. The series winner must win four games to be crowned World Champion. However, teams do not always play seven games. As soon as one team wins four games, the series is over. The following table lists the number of games played in every World Series from 1905 to 2011, along with the probability of that happening. For example, 20 times (19.4%) the series was over in four games. The probability table is given below.
- Calculate the expected number of games, the variance, and the standard deviation for the number of games played in a world series.
- Networks bid for World Series broadcasting rights, yet they cannot know with certainty how many games they will broadcast. Assume they don’t make a profit unless at least six games are played in the series. What is the probability they will make a profit in any given year?
3. The binomial tables used by the author are based on a cumulative distribution. Given a sample size (n) and an overall probability of success (p), you can solve probability problems for the binomial distribution using the tables. It is important to learn to use the tables because they help solve some problems much faster than using the binomial formula, which may require many calculations. The key to understanding the tables is that the table is based on a cumulative distribution. Sometimes you need to perform addition and subtraction to get the correct answer. Answer the following questions using the Binomial Table for n = 25 and p = 0.4.
- Solve the probability for x = 8 using the binomial formula.
- Next, get the same answer from the binomial tables. To use the table you have to subtract the cumulative probability for x = 7 from the cumulative probability for x = 8. This leaves the exact probability of 8. Confirm that the table calculation equals the formula calculation (to at least four decimal places).
- What is the probability of more than 8, P(x > 8)? To solve this, you need to subtract the table value for 8 from 1.0000.
- What is the probability of 4 or less, P(x < 5)?
- What is the probability of between 6 and 9, P(x > 5 and x < 10)?
4. Supposed an estimated 30% of high school students smoke cigarettes. You randomly select 25 high school students and survey them about their attitudes on smoking, and you ask them if they smoke. Look at the probabilities associated with the number of smokers out of 25.This is a binomial probability distribution problem with n = 25 and p = 0.30.
- What is the probability that at least 8 students smoke? P(x ≥ 8)?
- What is the probability that exactly 5 students smoke? P(x = 5)?
- Calculate the mean, variance, and standard deviation for this problem.
5. The standard normal table used by the author is based on μ = 0 and σ = 1. The table shows the probability from the center of the distribution out so many standard deviations in the right side of the curve. The right side of the distribution has a total probability of 0.5 (as does the left side). Given any mean and standard deviation from a normal distribution, computing a z-score converts the distribution to a standard normal. Since the normal distribution is symmetrical, the left side is a mirror image of the right side, and you can use the absolute value of a negative z-score to find probabilities on the left side of the curve. It is important to learn to use the standard normal table to solve normal distribution problems.
To get started, find the probabilities based on a normal distribution with a mean of 100 and a standard deviation of 10; that is, μ = 100 and σ = 10.
- The probability between 100 and 115
- The probability greater than 115
- The probability between 120 and 125
- The probability less than 175
- The probability greater than 181