Probability

This post is about some solved probability questions. These questions make use of (i) the Addition Law of Probabilities, and (ii) the Multiplication Law of Probabilities.

Solved Probability Questions

Question 1: Box A contains 5 Green and 7 Red balls. Box B contains 3 Green, 3 Red, and 6 Yellow balls. A box is selected at random, and a ball is drawn at random from it. What is the probability that the bill drawn is green?

Solution:

Box A

Total Balls: 5 + 7 = 12
Prob(Green) = $\frac{3}{12}$

Box B

Total Balls: 3 + 3 + 6 = 12
P(Green) = $\frac{3}{12} = \frac{1}{4}$

$$P(A+B) = P(A) + P(B) = \frac{5}{12} + \frac{3}{12} = \frac{8}{12} = \frac{2}{3}$$

Question 2: A pair of fair dice is thrown twice. What is the probability of getting a total of 5 or 11?

Solution:

\begin{align*}
P(X = 11 \,\, or X = 5) &= P(X=11) + P(X=15) – P(X=11\,\,and\,\, X=5)\\
P(X=11) &= \frac{2}{36}\\
P(X=5) &= \frac{4}{36}=\frac{1}{9}\\
P(X=11\,\, and X=5) &= 0
\end{align*}

Therefore,

\begin{align*}
P(X=11\,\, or X=5) &= P(X=11) + P(X=5) \\
&=\frac{2}{36} + \frac{1}{9} = \frac{1}{6}
\end{align*}

Note that $P(X=11\,\, and X=5) = 0$, because the sum of two dice cannot be at the same time 5 and 11.

Question 3: A marble is drawn at random from a box containing 10 red, 30 white, 20 blue, and 15 orange marbles. What is the probability that it is (i) orange or red (ii) not red or blue (iii) not blue, (iv) white, (v) red, white, or blue.

Solution:

Total number of balls = 10 + 30 + 20 + 15 = 75
Number of Orange balls = 15
Number of Blue balls = 20
Number of White balls = 30
Number of Red balls = 10

1. P(a marble drawn is red or orange) = P(Red marble) + P(Orange marble)
   $$=\frac{10}{75} + \frac{15}{75} = \frac{1}{3}$$
2. P(a marble drawn is not red or blue) = P(not Red) + P(Blue) – P(Blue and not Red)
   $$=\frac{65}{75} + \frac{20}{75} – \frac{20}{75} = \frac{65}{75}$$
3. P(a ball drawn is not Blue) = $1 – P(Blue) = 1 – \frac{20}{75} = 0.733$
4. P(a ball drawn is white) = $\frac{30}{75}$
5. P(a ball drawn is Red, White, or Blue) = P(Red) + P(White) + P(Blue)
   $$=\frac{10}{75} + \frac{30}{75} + \frac{20}{75} = \frac{60}{75}$$

Question 4: If two dice are thrown what are the various total number of dots that may turn up? What are the probabilities of each of them? What is the probability that the number of dots will total at least four?

Solution:

When two dice are thrown together, the minimum total number of dots is 2 (1, 1), and the maximum dots possible are 12 (6, 6). Therefore

• Probability of 2 dots (1, 1) = $\frac{1}{36}$
• Probability of 3 dots {(2, 1), (1, 2)} = $\frac{2}{36} = \frac{1}{18}$
• Probability of 4 dots {(2,2) (3,1) (1,3)} = $\frac{3}{36} = \frac{1}{12}$
• Probability of 5 dots {(4,1) (1,4) (2,3) (3,2)} = $\frac{4}{36} = \frac{1}{9}$
• {Probability of 6 dots {(3,3) (4,2) (2,4) (5,1) (5,1)} = $\frac{5}{36}$
• Probability of 7 dots {(4,3) (3,4) (5,2) (2,5) (6,1) (1,6)} = $\frac{6}{36} = \frac{1}{6}$
• Probability of 8 dots {(6,2) (2,6) (5,3) (3,5) (4,4)} = $\frac{5}{36}$
• Probability of 9 {(5,4) (4,5) (6,3) (3,6)} dots = $\frac{4}{36} = \frac{1}{9}$
• Probability of 10 dots {(5,5) (6,4) (4,6)} = $\frac{3}{36} = \frac{1}{2}$
• Probability of 11 dots {(5,6) (6,5)} = \frac{2}{36} = \frac{1}{18}$
• Probability of 12 dots {(6,6)} = $\frac{1}{36}$
• Probability that the number of dots will total at least 4 = $\frac{33}{36}$

Question 5: A one card is selected at random from a deck of 52 playing cards. What is the probability that the card is a club or a face card or both?

Solution:

\begin{align*}
P(club\,\, or\,\, face\,\, or\,\, both) &= P(club) + P(face) – P(club\,\, and\,\, face)\\
&=\frac{13}{52} + \frac{12}{52} – \frac{3}{52} = \frac{11}{26}
\end{align*}

Question 6: A class contains 10 men and 20 women of which half men and half women have brown eyes. What is the probability that a person chosen at random is a man or has brown eyes?

Solution:

Let $A$ be the event that it is a man (10 out of 30)
Let $B$ be the event that the person has brown eyes (5 men and 10 women: 15 out of 30)

$P(A\cap B)$ is a man AND has brown eyes $\frac{5}{30}$

\begin{align*}
P(A \cup B) &= P(A) + P(B) – P(A \cap B)\\
&= \frac{10}{30} + \frac{15}{30} – \frac{5}{30} = \frac{2}{3}
\end{align*}

Question 7: A drawer contains 50 bolts and 150 nuts. Half of the bolts and half of the nuts are rested. If one item is chosen at random, what is the probability that it is rusted or is a bolt?

Solution:

Number of Bolts = 50
NUmber of Nuts = 150
Total number of Items = 50 + 150 = 200

Item chosen is rusted: $P(A) = \frac{100}{200} = \frac{1}{2}$
Item chosen is bolt: $P(B) = \frac{50}{200} = \frac{1}{4}$
Ite is Rusted and Bolt = $P(A\cap B) = P(A) \cdot P(B) = \frac{1}{2}\cdot \frac{1}{4} = \frac{1}{8}$

\begin{align*}
P(A \cup B) &= P(A) + P(B) – P(A\cap B) \\
&= \frac{1}{2} + \frac{1}{4} – \frac{1}{8} = \frac{5}{8}
\end{align*}

Learn R Programming, Computer MCQs Online Test

The post is about Probability Quiz Online. There are 20 multiple-choice questions covering topics related to events and types of events, laws of probability, dependent and independent events, sample space, and probabilities related to coins, dice, and standard deck of cards. Let us start with Probability Quiz Online.

MCQs Probability Quiz Online

• The probability of the occurrence of the event ‘$A$’ is $P(A)=$
• When the occurrence of an event does affect the probability of the occurrence of another event it is called
• The probability of a sample space is equal to
• If three coins are tossed, the all possible cases are
• If a fair dice is rolled, the sample space is
• A fair dice is rolled twice, and the probability of getting a sum 8 is
• If $A$ and $B$ are mutually exclusive events then $P(A \cup B)=$
• If $A$ and $B$ are not mutually exclusive events then $P(A\cup B)=$
• If $A$ and $B$ are independent events then $P(A\cap B)=$
• If $A$ and $B$ are dependent events then $P(A \cap B) =$
• The probability of drawing a picture card from a pack of 52 cards is
• The probability of drawing a diamond card from a pack of 52 cards is
• The probability of drawing a ball at random from the box is
• For two mutually exclusive events $A$ and $B$, $P(A) = 0.3$ and $P(B)=0.5$ then $P(A \cup B)$ is
• If $P(B|A)=0.30$ and $P(A \cap B)=0.12$ then $P(A)$ is
• The probability of an event $A$ lies between
• If $P(A \cap B) = \phi$ then $P(A \cup B)=$ ————-.
• When an event is certain to occur, its probability is
• Baye’s Theorem
• The term “Sample Space” is used for

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The post is a quick overview of probability. Probability theory is a branch of mathematics that deals with the occurrence of random events. It provides a framework for quantifying uncertainty and making predictions based on available information.

Overview of Probability

The probability can be classified into two points of view:

• Objective/ Numerical Probability
• Subjective Probability

Objective probability requires a computational formula, while subject probability can be derived from experience, judgment, or previous knowledge about the event. In this post, I will focus on an overview of probability and the Laws of Probability.

Objective Probability

The objective probability has the following definitions

Classical and a Priori Definition

$$P(A) = \frac{\text{Number of sample points in the event based on the favorable events}}{\text{Number of sample points in the sample space}} = \frac{m}{n} = \frac{n(A)}{n(S)}$$

The Relative Frequency or a Posteriori Definition

$$P(A) = \lim\limits_{n\rightarrow \infty} \frac{m}{n}$$

This definition assumes that as $n$ increases to infinity, $\frac{m}{n}$ becomes stable.

When we experiment with the same condition many times, the probability of favourable event becomes stable. For example, if we toss a coin 10 times, then 100 times, then 1000 times, then 10,000 times, then 100,000 times, then 1000,000, and so on. We are also interested in the various numbers of heads that occur. Let $H$ (occurrence of head) be our favorable event, and the probability of a favorable event is called the probability of success. Then the definition said that there are approximately 50% heads in one million tosses. This definition is also called the empirical or statistical definition of probability. that is more useful in practical problems. In practical problems, we find the winning percentage of a team.

The axiomatic Definition of Probability

An axiom is a statement, about any phenomenon, which is used to find real-world problems.

The axiomatic definition of probability states that if a sample space $S$ with sample points $E_1, E_2, \cdots, E_n$, then a real number is assigned to each sample point denoted by $P(E_i)$, should satisfy the following conditions:

• for any event ($E_i$), $0< P(E_i) <1$
• $P(S) = 1$, sure event
• If $A$ and $B$ are two mutually exclusive events, then $P(A\cup B) = P(A) + P(B)$

Laws of Probability

For computing the probability of two or more events, the following laws of probability may be used.

Law of Addition

• For mutually exclusive events: $P(A\cup B) = P(A) + P(B)$
• For non-mutually exclusive events: $P(A\cup B) = P(A) + P(B) – P(A\cap B)$

If $A$, $B$, and $C$ are three events in a sample space $S$, then

$P(A\cup B \cup C) = P(A) + P(B) + P(C) – P(A \cap B) – P(B\cap C) – P(A \cap C)$

Law of Multiplication

For independent events $A$ and $B$: $P(A \text{ and } B) = P(A) \times P(B)$

For dependent events $A$ and $B$: $P(A \text { and } B) = P(A) \times P(B|A)$ (where $P(B|A)$ is the conditional probability of $B$ given $A$)

Law of Complementation

If $A$ is an event and $A’$ is the complement of that event, then

$P(A’) = 1-P(A)$, Note that $P(A) + P(A’) = 1$

Probability of sub-event

If $A$ and $B$ are two events in such a way that $A \subset B$, then $P(A) \le P(B)$

If $A$ and $B$ are any two events defined in a sample space $S$, then

$P(A\cap B’) = P(A) – P(A\cap B)$

Conditional Probability

$P(A|B) = \frac{P(A\cap B}{P(B)}$ or $P(B|A) = \frac{P(A\cap B}{P(A)}$.

Example of Conditional Probability

If we throw a die, what is the probability of 6? That is, $\frac{1}{6}$. What is the probability of 6 given that all are even numbers?

When a die is rolled, the sample space is $S=\{1, 2, 3, 4, 5, 6\}$. Let denote the even numbers by $B$, that is, $B=\{2, 4, 6\}$

$P(A|B) = \frac{1}{3}$

Law of Total Probability

If events $B_1, B_2, \cdots, B_n$ are mutually exclusive and exhaustive events, then for any event $A$: $P(A) = P(A|B_1) \times P(B_1) + P(A|B_2) * P(B_2) + \cdots + P(A|B_n) \times P(B_n)$

Bays’s theorem

Bays’ there is used to update probabilities based on new information.

If $A_1, A_2, \cdots, A_k$ are many events in a sample space.

$P(A_i|B) = \frac{P(A_i) P(B|A_i)}{\Sigma P(A)_i P(B|A_i)}, \text{ for } i, 1, 2, 3, \cdots, k$

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The post is about the Online MCQs Probability Quiz. There are 20 multiple-choice questions covering topics related to random experiments, random variables, expectations, rules of probability, events and types of events, and sample space. Let us start with the Probability Quiz.

Please go to MCQs Probability Quiz 10 to view the test

Online MCQs Probability Quiz with Answers

• Consider a dice with the property that the probability of a face with $n$ dots showing up is proportional to $n$. What is the probability of the face showing 4 dots?
• Let $X$ be a random variable with a probability distribution function $$f (x) = \begin{cases} 0.2 & \text{for  } |x|<1 \ 0.1 & \text{for } 1 < |x| < 4\ 0 & \text{otherwise} \end{cases}$$ The probability P (0.5 < x < 5) is ————-
• Runs scored by batsmen in 5 one day matches are 50, 70, 82, 93, and 20. The standard deviation is ————-.
• Find the median and mode of the messages received on 9 consecutive days 15, 11, 9, 5, 18, 4, 15, 13, 17.
• $E (XY)=E (X)E (Y)$ if $x$ and $y$ are independent.
• Mode is the value of $x$ where $f(x)$ is a maximum if $X$ is continuous.
• A coin is tossed up 4 times. The probability that tails turn up in 3 cases is ————–.
• If $E$ denotes the expectation the variance of a random variable $X$ is denoted as?
• $X$ is a variate between 0 and 3. The value of $E(X^2)$ is ————-.
• The random variables $X$ and $Y$ have variances of 0.2 and 0.5, respectively. Let $Z= 5X-2Y$. The variance of $Z$ is?
• In a random experiment, observations of a random variable are classified as
• A number of individuals arriving at the boarding counter at an airport is an example of
• If $A$ and $B$ are independent, $P(A) = 0.45$ and $P(B) = 0.20$ then $P(A \cup B)$
• If a fair dice is rolled twice, the probability of getting doublet is
• If a fair coin is tossed 4 times, the probability of getting at least 2 heads is
• If $P(B) \ne 0$ then $P(A|B) = $
• The collection of all possible outcomes of an experiment is called
• An event consisting of one sample point is called
• An event consisting of more than one sample point is called
• When the occurrence of an event does not affect the probability of occurrence of another event, it is called

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