Probability Theory Practice Questions

Probability theory is a branch of mathematics that studies random events and uncertainty. It provides methods to calculate the likelihood of different outcomes and helps analyze situations involving chance.

Question 1: Let's take two random dice and roll them randomly. Now the probability of getting a total of 10 is calculated.

Total Possible events that can occur (sample space) {(1,1), (1,2),..., (1,6),..., (6,6)}. The total spaces are 36.
Now the required events, {(4,6), (5,5), (6,4)} are all which adds up to 10.

So the probability of getting a total of 10 is = 3/36 = 1/12.

Question 2: A fair coin is tossed three times. What is the probability of getting exactly two heads?

Total possible outcomes when tossing a coin three times = 2³ = 8.
Possible outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT.
Outcomes with exactly two heads: HHT, HTH, THH (3 outcomes).

Probability of getting exactly two heads:
P(exactly 2 heads)=Number of favorable outcomes/ Total outcomes.

P(exactly 2 heads)=3/ 8.

Question 3: A standard deck of cards contains 52 cards. What is the probability of drawing an Ace or a King from the deck?

Total number of cards = 52.
Number of Aces = 4.
Number of Kings = 4.

Total number of favorable outcomes (Aces or Kings) = 4 + 4 = 8.

Probability of drawing an Ace or a King:

P(Ace or King)=Number of favorable outcomes/Total outcomes
P(Ace or King)=Number of Aces or Kings/Total number of cards.

P(Ace or King) = 8/52 = 2/13.

Question 4: Consider a jar with 7 red marbles, 3 green marbles and 4 blue marbles. What is the probability of randomly selecting a non-blue marble from the jar?

Given,
Number of Red Marbles = 7, Number of Green Marbles = 3, Number of Blue Marbles = 4

So, Total number of possible outcomes in this case: 7 + 3 + 4 = 14
Now, Number of non-blue marbles are: 7 + 3 = 10
According to the formula of theoretical Probability we can find, P(Non-Blue) = 10/14 = 5/7 

Hence, theoretical probability of selecting a non-blue marble is 5/7. 

Question 5: Consider for players Naveena and Isha, playing a table tennis match. The probability of Naveena winning the match is 0.76. What is the probability of Isha winning the match?

Let N and M represent the events that Naveena wins the match and Isha wins the match, respectively. 

The probability of Naveena’s winning P(N) = 0.76(given)
The probability of Isha's winning P(I) = ?
Winning of the match is an mutually exclusive event, since only one of them can win the match. 
Therefore, 

P(N) + P(I) =1 
P(I) = 1 –  P(N) 
P(I) = 1 –  0.76 = 0.24

Thus, the Probability of Isha winning the match is 0.24.

Question 6: If someone takes out one card from a 52-card deck, what is the probability of the card being a heart? What is the probability of obtaining a 7-number card? 

Total number of cards in a deck = 52
Total Number of heart cards in a deck = 13

So, the probability of obtaining a heart, 

P(heart) = 13/52 = 1/4
Total number of 7-number cards in a deck = 4

So, the probability of obtaining a 7-number card, 
P(7-number) = 4/52 = 1/13.

Practice Questions

Question 1: Two fair dice are rolled at the same time. What is the probability of getting a total of 8?

Question 2: A fair coin is tossed four times. What is the probability of getting exactly three heads?

Question 3: One card is drawn from a standard 52-card deck .What is the probability of drawing a Queen or a Heart?

Question 4: A jar contains 5 red balls, 6 blue balls, and 9 green balls. What is the probability of selecting a green ball at random?

Question 5: The probability that a student passes an exam is 0.82. What is the probability that the student fails the exam?