Probability

The post contains MCQs Probability Questions with Answers. There are 20 multiple-choice questions covering topics related to the statistical experiment, basics of probability, sample space, addition rule of probability, multiplication rule of probability, and conditional probability. Let us start with MCQs Probability Questions.

MCQs Probability Questions with Answers

• The Complement of $P(A|B)$ is
• The probability of an intersection of two events is computed by using the
• If two events $A$ and $B$ are mutually exclusive events, then
• The range of probability is
• In a statistical experiment, each time the experiment is repeated
• The set of all possible outcomes (sample points) is called
• The sample space (experimental outcomes) refers to
• An experiment that consists of tossing 4 coins successively. The number of sample points in this experiment is
• On a December day, the probability of snow is 0.30. The probability of a “cold” day is 0.50. The probability of snow and a “cold” is 0.15. Do snow and “cold” weather are independent events?
• If $P(A)=0.5$ and $P(B)=0.5$, then $P(A \cap B)$ is
• If $A$ and $B$ are independent events with $P(A)=0.6$ and $P(B)=0.6$, then $P(A \cap B)=$?
• If events $A$ and $B$ are independent events with $P(A)=0.2$ and $P(B)=0.6$, then $P(A \cup B)=$?
• If $A$ and $B$ are independent events with $P(A)=0.4$ and $P(B)=0.25$, then $P(A \cup B)=$?
• Events $A$ and $B$ are mutually exclusive. Which of the following statements is true?
• If events $A$ and $B$ are independent events with $P(A)=0.05$ and $P(B)=0.65$, then $P(A|B)=$?
• A six-sided die is tossed three times. The probability of observing three ones in a row is
• If $P(A|B)=0.3$
• If events $A$ and $B$ are independent events with $P(A)=0.1$ and $P(B)=0.4$, then
• If $P(A|B)=0.3$ and $P(B)=0.8$, then
• If $P(A)=0.6$, $P(B)=0.3$, and $P(A \cap B)=0.2$, then $P(B|A)=$?

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Introduction to Empirical Probability

An empirical probability (also called experimental probability) is calculated by collecting data from past trials of the experiments. The experimental probability obtained is used to predict the future likelihood of the event occurring.

Formula and Examples Empirical/ Experimental Probability

To calculate an empirical/ experimental probability, one can use the formula

$$P(A)=\frac{\text{Number of trials in which $A$ occurs} }{$\text{Total number of trials}}$$

• Coin Flip: Let us flip a coin 200 times and get heads 105 times. The empirical probability of getting heads is $\frac{105}{200} = 0.525%, or 52.5%.
• Weather Prediction: Let you track the weather for a month and see that it rained 12 out of 30 days. The empirical probability of rain on a given day that month is $\frac{12}{30} = 0.4$ or 40%.
• Plant Growth: Let you plant 50 seeds and 35 sprout into seedlings. The experimental probability of a seed sprouting is $\frac{35}{50} = 0.70$ or 70%.
• Board Game: Suppose you play a new board game 10 times and win 6 times. The empirical probability of winning the game is $\frac{6}{10} = 0.6$ or 60%.
• Customer Preferences: In a survey of 100 customers, 80 prefer chocolate chip cookies over oatmeal raisins. The empirical probability of a customer preferring chocolate chip cookies is $\frac{80}{100} = 0.80$ or 80%.
• Basketball Game: A basketball player practices free throws and makes 18 out of 25 attempts. The experimental probability of the player making their next free throw is $\frac{18}{25} = 0.72$ or 72%.

Empirical Probability From Frequency Tables

A frequency table calculates the probability that a certain data value falls into any data group or class. Consider the frequency table of examination scores in a certain class.

Class	Frequency ($f$)	$frf$
40 – 49	1	$\frac{1}{20}=0.05$
50 – 59	2	$\frac{1}{20}=0.10$
60 – 69	3	$\frac{3}{20}=0.15$
70 – 79	4	$\frac{4}{20}=0.20$
80 – 89	6	$\frac{6}{20}=0.30$
90 – 99	4	$\frac{4}{20}=0.20$

Let event $A$ be the event that a student scores between 90 and 99 on the exam, then

$$P(A) = \frac{\text{Number of students scoring 90-99}}{\text{Total number of students}} = \frac{4}{20} = 0.20$$

Notice that $P(A)$ is the relative frequency of the class 90-99.

Key Points Empirical/ Experimental Probability

• It is based on actual data, not theoretical models.
• It is a good approach when the data is from similar events in the past.
• The more data you have, the more accurate the estimate will be.
• It is not always perfect, as past results do not guarantee future outcomes.

Limitations Empirical/ Experimental Probability

• It can be time-consuming and expensive to collect enough data.
• It may not be representative of the future, especially if the underlying conditions change.

FAQS about Empirical/ Experimental Probability

1. Define empirical probability.
2. How one can compute empirical probability, write the formula of empirical probability.
3. Give real-life examples of empirical/ experimental probability.
4. What are the limitations of empirical/ experimental probability?
5. How does empirical/ experimental probability resemble with frequency distribution, explain.

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Online MCQs Probability Questions and Answers. The Quiz covers topics of rules of counting, events, and types of events such as mutually exclusive and exhaustive events, sample space, Rules of Probability, etc. Let us start with the MCQs Probability Questions and Answers.

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Probability Questions and Answers

• In tossing two dice, the probability of obtaining 4 as the sum of the resultant faces is ———-.
• The probability of a red card out of 52 cards is
• A single letter is selected at random from the word “STATISTICS” the probability that it is a vowel is ———-.
• A single letter is selected at random from the word “PROBABILITY”, the probability that it is a vowel is ———.
• When two dice are rolled, the probability of getting similar faces.
• $P(A)=0.2$, $P(B)=0.5$. If $A, B$, and $C$ are mutually exclusive events then $P(C)$ is
• $P(A)=0.7$, $P(B)=0.5$. If $A$ and $B$ are independent then $P(A \cap B)=$?
• For two independent events $A$ and $B$, if $P(A)=0.12$, $P(B)=0.2$ then $P(A\cap B)=$?
• If $P(A)=0.3$, $P(B)=0.8$, $P(A\cap B)=0.24$ then the events $A$ and $B$ are
• If $P(A \cup B)=0.8$, $P(A)=0.4$, $P(A\cap B)=0.3$ then the values of $P(C)$ is
• The probability of drawing any one spade card is
• The probability of drawing a white ball from a bag of 6 red, 8 black, 10 green, and 5 white balls is
• A student solved 160 questions out of 350. The probability of solving the next is
• If $E$ is an impossible event, then $P(E)$ is
• If the event contains no number in it, then the probability of such event will be
• If $P(B)\ne 0$, then the conditional probability $P(A|B)$ is defined a
• If $A$ and $B$ are two independent events then $P(A)\dot P(B)=$
• $P(A\cup B)=P(A) + P(B)$, if $A$ and $B$ are
• If $A$ and $B$ are two non-mutually exclusive events then $P(A\cup B)=$
• $P(A\cap B)=P(A)\dot P(B)$ then events $A$ and $B$ are

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Online MCQs Probability Quiz Answers. The Quiz covers topics of rules of counting, events, and types of events such as mutually exclusive and exhaustive events, sample space, Rules of Probability, etc. Let us start with the MCQs Probability Quiz Answers.

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Online MCQs Probability Quiz Answers

• In how many ways can 6 persons be seated on a sofa set with three seats?
• The number of ways in which four books can be arranged on a shelf is
• Number of ways a committee of 3 members can be selected from 5 members
• $^nC_r=$
• $^5P_1=$
• Two events are called collectively exhaustive if $A\cup B=$
• If $A \cap B=\phi$, then the events $A$ and $B$ are called
• When a coin is tossed, the sample space is
• A coin is tossed three times in succession the number of sample points in the sample space is
• Three coins are tossed together, the sample will consist of ———- sample points.
• When a die and coin are rolled there are sample points.
• When a pair of dice is rolled, the sample space consists of
• Total number of ways when three fair dice are thrown
• If three cards are drawn from a pack of 52 cards, then sample space is
• The probability of an event always lies between
• The probability of an event cannot be
• The sum of probabilities of all mutually exclusive events of an experiment will be
• In tossing two perfect coins the probability that at least one head will occur is
• If a coin is tossed thrice then the probability of three heads is
• Three coins are tossed, ———- is the probability of getting at least one head.

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