Sample Problems on Events in Probability

Here we have provided you with a few solved sample problems on events in probability:

Problem 1: A die is thrown in the game of Ludo and E₁ denotes the event of getting even numbers and E₂ represents the event of getting a number more than 3, Find the Set for the following events,

1. E₁ or E₂
2. E₁ and E₂

Solution:

The sample space for the die will be,

S= {1, 2, 3, 4, 5, 6}

• E₁ (only even numbers) = {2, 4, 6}
• E₂ (number more than 3) = {4, 5, 6}

1. E₁ or E₂ = {2, 4, 5, 6}
2. E₁ and E₂ = {4, 6}

Problem 2: A die is thrown and the set for the sample space obtained is, S = {1, 2, 3, 4, 5, 6}

E₁ is defined as the event of obtaining a number less than 5 and E₂ is defined as the event of obtaining a number more than 2.

Find the set for the following,

1. E₁ but not E₂
2. E₂ but not E₁

Solution:

Sample space will be, 

S= {1, 2, 3, 4, 5, 6}

• E₁ (a number less than 5)= {1, 2, 3, 4}
• E₂ (a number more than 2)= {3, 4, 5, 6}

1. E₁ but not E₂ = {1, 2}
2. E₂ but not E₁ = {5, 6}

Problem 3: Write the sample space for tossing three coins at once, also answer the event of 2 exactly 2 heads at a time.

Solution:

Tossing Three Coins the sample space is,

S = {(H, H, H), (H, H, T), (H, T, H), (T, H, H), (T, T, H), (T, H, T), (H, T, T), (T, T, T)}

Hence, the Sample Space Comprises 6 Possible Outcomes

Event (E) for the occurrence of exactly two heads,

• E = {(H, H, T), (H, T, H), (T, H, H)}

Problem 4: Name the types of events obtained from the given below experiments,

1. A coin is tossed for the 5th time and in the event of getting a tail when the first four times, the result was ahead.
2. S (sample space)= {1, 2, 3, 4, 5} and E= {4}
3. S= {1, 2, 3, 4, 5} and E= {2, 4}
4. S= {1, 2, 3, 4, 5}, E₁= {1, 2} and E₂= {3, 4}

Solution:

1. No matter how many times the coin is tossed, every time the probability of getting a tail will be 0.5 irrespective of the previous outcomes, therefore the event will be an independent event.
2. E= {4} is a Simple event.
3. E= {2, 4} is a compound event.
4. E₁ and E₂ are Mutually exclusive events.

Problem 5: Sample Space of an experiment is given as,

S = {10, 11, 12, 13, 14, 15, 16, 17} and the event, E is defined as all the even numbers. What will be the complementary event for E?

Solution:

S = {10, 11, 12, 13, 14, 15, 16, 17}

E (All even numbers) = {10, 12, 14, 16}

E’ (complementary of E) = {11, 13, 15, 17}

Problem 6: Consider the experiment of tossing a fair coin 3 times, Event A is defined as getting all tails. What kind of event is this? 

Solution: 

Sample space for the coin toss will be, 

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

For the event A, 

A = {TTT}

This event is only mapped to one element of sample space. Thus, it is a simple event. 

Problem 7: Let’s say a coin is tossed once, state whether the following statement is True or False. 

“If we define an event X which means getting both heads and tails. This event will be a simple event.”

Solution:

When a coin it tossed, there can be only two outcomes, Heads or Tails. 

S = {H, T} 

Getting both Heads and Tails is not possible, thus event X is an empty set. 

Thus, it is an impossible and sure event. So, this statement is False. 

Problem 8: A die is rolled, and three events A, B, and C are defined below:

• A: Getting a number greater than 3 
• B: Getting a number that is multiple of 3. 
• C: Getting an odd number

Find A ∩ B, A ∩ B ∩ C, and A ∪ B.

Solution:

Sample space for die roll will be, 

S = {1, 2, 3, 4, 5, 6} 

For the event A, 

A = {4, 5, 6}

For the event B, 

B = {3, 6}

For the event C, 

C = {1, 3, 5}

A ∩ B = {4, 5, 6} ∩ {3, 6}

⇒ A ∩ B = {6}

A ∩ B ∩ C = {4, 5, 6} ∩ {3, 6} ∩ {1, 3, 5}

⇒ A ∩ B ∩ C = ∅ (Empty Set) 

A ∪ B = {4, 5, 6} ∪ {3, 6}

⇒ A ∪ B = {3, 4, 5, 6}

Problem 9: A die is rolled, let’s define two events, event A is getting the number 2 and Event B is getting an even number. Are these events mutually exclusive? 

Solution: 

Sample space for die roll will be, 

S = {1, 2, 3, 4, 5, 6} 

For the event A, 

A = {2}

For the event B, 

B = {2, 4, 6}

For two events to be mutually exclusive, their intersection must be an empty set 

A ∩ B = {2} ∩ {2, 4, 6}

⇒ A ∩ B  = {2}

Since it is not an empty set, these events are not mutually exclusive.

Events in Probability- In Probability, an event can be defined as a set of outcomes of an experiment. In other words, an event in probability is the subset of the respective sample space.

In this article, we will learn about the Events in Probability, Types of Events in Probability, definitions, how they are classified, how the algebra of events works, etc.