Probability Examples and Explanations

You visit a friend who lives in the suburbs of Chicago. You decide to take a commuter train into the city. Your friend says that a train stops at the station every 30 minutes. Without any more information, you logically apply the uniform probability distribution and determine that you will wait between 0 and 30 minutes for a train with a probability of 1.00. You arrive at the train station and start timing your wait time. A train arrives 35 minutes later.
a. Given your friend’s information, what was the probability that a train arrives in 35 minutes or more?
b. Given your friend’s information, what conclusion can you make about your friend’s information?

What is the probability of producing a child that will phenotypically resemble either one of the two parents in the following four crosses? How many phenotypically different kinds of progeny could potentially result from each of the four crosses?
Aa Bb Cc Dd ${\displaystyle \times }$ Aa Bb Cc Dd

Cars arrive at a toll both according to a Poisson process with mean 80 cars per hour. If the attendant makes a one-minute phone call, what is the probability that at least 1 car arrives during the call?

Compute the probability of event E if the odds in favor of E are ${\displaystyle \frac{4}{1}}$

You are told that a coin approximately has a probability of 0.25 of landing
on heads if the probability that it lands on heads is between 0.23 and 0.27. You toss a coin
100,000 times and it lands on heads 30,000 times. Find a lower bound on the probability
that the probability is not approximately 0.25 that the coin land on heads.''

In a​ state's Pick 3 lottery​ game, you pay ​$1.23 to select a sequence of three digits​ (from 0 to​ 9), such as 266. If you select the same sequence of three digits that are​ drawn, you win and collect ​$382.15 Complete parts​ (a) through​ (e).
a. How many different selections are​ possible?
b. What is the probability of​ winning?
(Type an integer or a​ decimal.)
c. If you​ win, what is your net​ profit?
​(Type an integer or a​ decimal.)
d. Find the expected value.
​$(Round to the nearest hundredth as​ needed.)
e. If you bet $ 1.23 in a certain​ state's Pick 4​ game, the expected value is negative $ 0.85−. Which bet is​ better, a​$1.23bet in the Pick 3 game or a $ 1.23 bet in the Pick 4​ game? Explain.
A.
The Pick 3 game is a better bet because it has a larger expected value.
B.
Neither bet is better because both games have the same expected value.
C.
The Pick 4 game is a better bet because it has a larger expected value.

A standard 52 card deck has four 13 card suits: diamonds, hearts, clubs, and spades. The diamonds and hearts are red, and the clubs and spades are black. Each 13 card suit contains cards numbered from 2 to 10, a jack, a queen, a king, and an ace. An expierement consists of drawing 1 card from the deck. Find the probability of drawing a black ace of hearts.

Good Fortune supermarket’s manager of San Gabriel claims that the average time to clean 10 pounds of catfish is 8 minutes. A random sample of 4 employees shows that the average time is 9 mins with a standard deviation of 2 minute. At the 0.05 significance level, can we conclude that the mean time is 8 minutes? What is the P-value? Leave your answer in 4 decimal places.

The grades for the chapter 5 test are summarized below: Min = 5 Max = 100 Lower Quartile = 55 Upper Quartile = 85 Median = 75 N=74 Which of the following statements is true?a)Both 5 and 100 are outliers. It is possible that there are also other outliers. b)The largest observation is 100 and it is an outlier. No other observations in the data set could be outlersc)The smallest observation is 5 and it is an outlier. No other observations in the data set could be outliers.d)5 is an outlier and it is possible that there are other outliers at the low end of the data set. There are no outlers at the high end of the data e)100 is an outlier and it is possible that there are other outliers at the high end of the data set.

To play Florida lotto, one chooses six numbers from 1 to 53. What is the chance of winning FL lottery?

For the Illinois Lottery Pick 3 game, a players must match a sequence of three repeatable numbers, ranging from 0 to 9, in exact order. With a single ticket, what is the probability of matching the three winning numbers?

Winning numbers for the "Straight" DC-4 lottery game are reported on television and in local newspapers. You pay $1.00 and pick a four-digit number. The state chooses a four-digit number at random and pays you $5000 if your number is chosen. What are the expected winnings from a $1.00 Straight DC-4 wager? Please round your final answer to the nearest cent.

In a​ state's Pick 3 lottery​ game, you pay ​$1.27 to select a sequence of three digits​ (from 0 to​ 9), such as 244. If you select the same sequence of three digits that are​ drawn, you win and collect ​$468.14.
a. How many different selections are​ possible?
b. What is the probability of​ winning?
c. If you​ win, what is your net​ profit? ​
d. Find the expected value.
e. If you bet $1.27 in a certain​ state's Pick 4​ game, the expected value is negative −$0.80. Which bet is​ better, a ​$1.27 bet in the Pick 3 game or a $ 1.27 bet in the Pick 4​ game? Explain.
A. The Pick 4 game is a better bet because it has a larger expected value.
B. The Pick 3 game is a better bet because it has a larger expected value.
C. Neither bet is better because both games have the same expected value.

The average height of U.S. men is 69 inches (5 feet 9 inches) with a standard deviation of 4 inches. What is the z-value that corresponds to the shortest 33% of men?
Select one:
a. -0.95
b. -0.13
c. -0.33
d. -0.44
e. 0.95
f. 0.44
g. 0.33
h. 0.13

Assume that​ women's heights have a mean of 63 inches ​(5 feet 3​inches) and a standard deviation of 3 inches. a. What​ women's height corresponds to a​ z-score of 1.20​? b. What​ women's height corresponds to a​ z-score of negative −1.00​?

The masses of oil palm sold in container have a normal distribution with average of 1,000kg and standard deviation of 35kg. Find the probability that product will weight
a) Less than 978kg
b) Less than 1,027kg
c) Between 978kg and 1,027kg

PNC Bank agency receives in average 20 customers per hour. Due to the current capacity, this agency has an average service time of 30 minutes every hour. The manager would like you to perform the following analysis.
What is the probability of finding a busy system?
Expected number of customers in the system
Expected number of customers waiting in the system

What is the formula to find the number of permutations to 4 sets of 6 numbers?