Probability Examples and Explanations

Suppose our experiments consist ofrolling a six-sided die. Then we can define the appropriate sample space as ${\displaystyle S=\{1,2,3,4,5,6\}}$.
I.What is probability that we roll a 2?
II.What is probability that we roll a 2 or 5?
III.What is probability that we roll an even number?

The probability of a student cheating is 0.6, the probability of a student getting caught given that he/she has cheated is 0.3 and the probability of a student not getting caught given that he/she has cheated is 0.7. Lets

The probability that a shopper will buy a PS3 is 0.40.
The probability that a shopper will buy an X-Box 360 is 0.54.
The probability that a shopper will buy a Nintendo DS is 0.30.
The probability that a shopper will buy both a PS3 and an X-Box 360 is 0.13. The probability that a shopper will buy both a PS-3 and a DS is 0.12.
The probability that a shopper will buy both an X-Box 360 and a DS is 0.07. There is no chance that a shopper will buy all three systems.
Find the probability that a random shopper will:
a) Buy only a PS-3
b) Buy exactly one type of system
c) Buy exactly two types of systems d) Not buy any of the three systems

A standard deck of cards contains 52 cards. One card is selected from the deck.
a) Compute the probability of randomly selecting a spade or diamond.
b) Compute the probability of randomly selecting a spade or diamond or heart.
c) Compute the probability of randomly selecting a king or club.
a. ${\displaystyle P\left(\text{spade or diamond}\right)=?}$
b. ${\displaystyle P\left(\text{spade or diamond or heart}\right)=?}$
c. ${\displaystyle P\left(\text{king or club}\right)=?}$

Suppose that the random variable x, shown below, represents the number times . P(x) represents the probability of a randomly selected person having received that number of speeding tickets during that period. Use the probability distribution table shown below to answer the following questions.
$$\begin{array}{|cc|}\hline x& P(x)\\ 0& 0.3176\\ 1& 0.2705\\ 2& 0.2469\\ 3& 0.0832\\ 4& 0.0485\\ 5& 0.0333\\ 6+& 0.0000\\ \hline\end{array}$$
a) What is the probability that a randomly selected person has received four tickets in a three-year period?
${\displaystyle P(x=4)=P(x=4)=}$
b) What is the probability that a randomly selected person has received three tickets in a three-year period?
${\displaystyle P(x=3)=P(x=3)=}$
c) What is the probability that that a randomly selected person has received more than one tickets in a three-year period?
${\displaystyle P\left(x>1\right)=P\left(x>1\right)=}$
d) What is the probability that that a randomly selected person has received two or less tickets in a three-year period?
${\displaystyle P(x\le 2)=P(x\le 2)=}$

From Theoretical and Experimental Probability, which of the two method is more appropriate and valid to use in determining probabilities? Why?

Suppose your top drawer contains 44 different colored socks:
6 are white,
16 are black,
10 are pink, and
12 are blue.
All socks in the drawer are loose (unpaired). In the morning, you randomly select two socks, one at a time. Calculate the following probabilities, writing your answer either as a decimal or a fraction.
(a) What is the probability that you get a blue pair of socks?
(b) What is the probability that you do not get a blue pair of socks?
(c) What is the probability that neither is blue?
(d) What is the probability that you get either a white pair or a blue pair of socks?
(e) What is the probability that you get one black sock and one white sock?

If the probability that it rains on Thursday is 30% and the probability that it rains on Sunday is 20%, what is the probability that it does not rain on both days?

Tony and Sam play chess (either of them wins, draw is not allowed). If Tony plays White, probability of him winning is 0.72. If Tony plays Black, probability of him winning is 0.63. Tony and Sam play 2 games and in second game they change sides. Find the probability that Sam wins both games.

The probability of event ${\displaystyle A^c}$ is 0.4, the probability of ${\displaystyle {\left(AUB\right)}^c}$ is 0.1, and the probability of the intersection of Aand Bis.2. Whats the probability of event 8?

Assume that the business makes a profit with probability 0.4 in the first year. For each year thereafter, the business makes a profit with probability 0.6 if it made a profit in the previous year, and with probability 0.4 if it did not make a profit in the previous year.
What is the probability that the business makes a profit in exactly two of its first three years?

In shooting arena there are 5 different guns. Probability of hitting aim from gun1 is 0.5.
Probability of hitting aim from gun2 is 0.55. Probability of hitting aim from gun3 is 0.7.
Probability of hitting aim from gun4 is 0.75. Probability of hitting aim from gun5 is 0.4. One shoot was made from a gun selected at random. What is the probability that aim was hit?

Given the probability that "she's up all night 'til the sun" OR "she's up all night for good fun" is 0.31, the probability that "she's up all night for good fun" is 0.49, and the probability that "she's up all night 'til the sun" AND "she's up all night for good fun" is 0.39, what's the probability that "she's up all night 'til the sun"?

The table shows the results of a survey of 100 authors by a publishing company.
$$\begin{array}{|cccc|}\hline & \text{New Authors}& \text{Established Authors}& Total\\ Successful& 15& 30& 45\\ Unsuccessful& 20& 35& 55\\ Total& 35& 65& 100\\ \hline\end{array}$$
Compute the following conditional probability. (Enter your probability as a fraction.)
An author is established, given that he is unsuccessful.

Two cards are drawn without replacement from an ordinary​ deck, find the probability that the second is a face card​, given that the first is not a face card.
What is the conditional​ probability?
nothing ​(Type an integer or a simplified​ fraction.)
The Motor Vehicle Department has found that the probability of a person passing the test for a​ drivers

The probability of someone getting covid positive is 0.6, the probability of someone dying suddenly given that
he/she is covid positive is 0.3 and the probability of someone dying suddenly given that he/she is not covid
positive is 0.1. Lets

A Geiger counter counts the number of alpha particles from radioactive material. Over a long period of time, an average of 15 particles per minute occurs. Assume the arrival of particles at the counter follows a Poisson distribution.
a) Find the probability of exactly 20 particles arrive in a particular one minute period.
b) Find the probability of exactly one particle arrives in a particular one second period.
c) Find the probability that at least one particle arrives in a particular one second period.
d) Find the probability that at least two particles arrive in a particular 4 second period.

The probability that a regulary seheduted flightdeparts on time is 0.83 ,
the probability that it arrives on time is 0.92 and the probability that it departs
and arrives on time is 0.78 . Find the probability thata plane
(1) arrives on time given thatit departed on time .
(2) departed on time given that it has arrived on time .