Probability Examples and Explanations

Consider a Poisson distribution with a mean of two occurrences per time period.
Write the appropriate Poisson probability function.

Consider a Poisson distribution with a mean of two occurrences per time period.
What is the expected number of occurrences in three time periods?

Martha invited 4 friends to go with her to the movies. There are 120 different ways in which they can sit together in a row of 5 seats, one person per seat. In how many of those ways is Martha sitting in the middle seat?

Uniform distribution is defined over the interval from 2 to 5.
Find the probability of a value more than 2.6.

Consider a Poisson distribution with a mean of two occurrences per time period.
Compute the probability of two occurrences in one time period.

Let X be a random variable with range [0,1] and probability density function (pdf) is given as:
${\displaystyle f\left(X\right)=3x^2}$. Find probability ${\displaystyle P\left(X>\frac{1}{5}\right)}$.

Suppose George wins 38​% of all thumb wars.
​(a) What is the probability that George wins two thumb wars in a​ row?
​(b) What is the probability that George wins four thumb wars in a​ row?
​(c) When events are​ independent, their complements are independent as well. Use this result to determine the probability that George wins four thumb wars in a​ row, but does not win five in a row.

Assume that the coins are fair. If a sample point is the result of n tosses, assign a probability of ${\displaystyle {\left(\frac{1}{2}\right)}^n}$ to that sample point. Verify that the probability of the whole sample space is 1. Calculate the probability that the second toss results in heads.

1. Define theoretical probability.
2. What is the probability of each of the following?
a) picking the 9 of clubs from a deck of cards ? ${\displaystyle \left\{\frac{t}{i}on\right\}}$
b) flipping heads with a coin ? ${\displaystyle \left\{decimal\right\}}$
c) picking a diamond from a deck of cards ? ${\displaystyle \left\{percent\right\}}$
d) rolling a with 1 die ? ${\displaystyle \left\{\frac{t}{i}on\right\}}$
e) rolling an even number with 1 die ? ${\displaystyle \left\{decimal\right\}}$
f) flipping heads for or tails with a coin ? ${\displaystyle \left\{percent\right\}}$
3. Define experimental probability.

If two dice are rolled , find the probability that neither die shows a 2 on it .
probability=

A shipment of 30 compressors contains 5 defectives. if we select a sample of 5 units at random for inspection:
a) what is the probability that it contains exactly one defective?
b) what is the probability that it contains at least one defective?
c) what is the probability that it contains at least two defective?

The next number in the series 38, 36, 30, 28, 22 is ?

Suppose a die is weighted such that the probability of rolling a three is the same as rolling a six, the probability of rolling a one, two, or four is 3 times that of six, and the probability of rolling a five is 5 times that of rolling a three. Find the probability of 1. rolling a one 2. rolling a two 3. rolling a three 4. rolling a four 5. rolling a five 6. rolling a six

Probability of an event are given, compute the probability of the event (enter the probability as a fraction)
2 to 7

An automobile manufacturer produces a certain model of car. The fuel economy figures of these cars are normally distributed with a mean mileage per gallon (mpg) of 36.8 and a standard deviation of 1.3
(a) What is the probability that one of these cars will have an mpg of more than 37.5?
(b) What is the probability that one of these cars will have an mpg of less than 35?
(c) What is the probability that one of these cars will have an mpg of less than 40?

According to the Chronicles of Contrived Statistics (March, 2016), the probability that Elon Musk will offer you a free trip into space in the next month is 49%. The probability that scientists from Area 51 will start selling pet aliens next month is 46%. Finally, the probability that either Elon Musk will offer you a free trip to space or that scientists will begin sales of pet aliens next month is 31%. What is the probability that both Elon Musk will offer you a free trip into space and scientists from Area 51 will start selling pet aliens in the next month?

Tendulkar is playing a cricket tournament in which each player can play 200 balls and the player who hits maximum sixes wins. Tendulkar hits a six on the first ball, but misses out on the second.
After that, the probability that he hits a six is equal the proportion of sixes that he has hit so far. For example, at the third ball, the probability is ${\displaystyle \frac{1}{2}}$.
(a) What is the probability that Tendulkar hits only one six?
(b) What is the probability that Tendulkar hits 100 sixes?

a) Mr Okyere takes banana and apple to the office to share with his office mate. The probability that he will eat the banana is 0.7 and 0.8 is the probability that he will eat the apple. The probability that he will eat both fruits is 0.6. Find the conditional probability that he will eat the apple given that he does not eat the banana.
b) The Ministry of Agriculture in its fight against the army worm infection of maize farms in the country has approved three brands of chemicals A, B and C to be used by farmers as follows: Farmers are to combine exactly two of the chemicals to spray their farms or they should use none of them. The proportions of farmers who opted for chemicals A, B and C are respectively given by i ${\displaystyle \frac{1}{5},\frac{1}{3}}$ and ${\displaystyle \frac{7}{15}}$ A farmer is randomly selected from the group of farmers. What is the probability that the farmer chose none of the chemicals?
c) Consider three boxes containing a brand of light bulbs. Box I contains 6 bulbs of which 2 are defective, Box 2 has 1 defective and 3 functional bulbs and Box 3 contains 3 defective and 4 functional bulbs. A box is selected at random and a bulb drawn from it at random is found to be defective. Find the probability that the box selected was Box 2.