Event A: The outcome of the rolled die is a prime number.
Event B: The outcome of the rolled die is an odd number less than 4.
The conditional probability of A given B, denoted by P(A|B), is ______ (rounded off to one decimal place).
Die Roll Experiment: Probability Calculation
We need to find the conditional probability P(A|B) for rolling an unbiased die once.
Sample Space and Events
- The sample space (S) for rolling an unbiased die is S = {1, 2, 3, 4, 5, 6}. The total number of possible outcomes is |S| = 6.
- Event A is defined as the outcome being a prime number. The prime numbers in S are {2, 3, 5}. So, A = {2, 3, 5}.
- Event B is defined as the outcome being an odd number less than 4. The odd numbers in S less than 4 are {1, 3}. So, B = {1, 3}.
Intersection of Events
The intersection of events A and B (A ∩ B) represents outcomes that are both prime and odd numbers less than 4. From the sets A={2, 3, 5} and B={1, 3}, the common outcome is {3}.
- A ∩ B = {3}
- The number of outcomes in the intersection is |A ∩ B| = 1.
Calculating Probabilities
The probability of an event is the number of favorable outcomes divided by the total number of outcomes.
- Probability of the intersection:
P(A ∩ B) = \frac{|A \cap B|}{|S|} = \frac{1}{6}
- Probability of Event B:
P(B) = \frac{|B|}{|S|} = \frac{2}{6} = \frac{1}{3}
Conditional Probability P(A|B)
The formula for conditional probability is P(A|B) = \frac{P(A \cap B)}{P(B)}.
- Substitute the calculated probabilities:
P(A|B) = \frac{1/6}{1/3}
- Simplify the expression:
P(A|B) = \frac{1}{6} \times \frac{3}{1} = \frac{3}{6} = \frac{1}{2}
Final Result
The conditional probability P(A|B) is 0.5.
The question asks for the value rounded off to one decimal place, which is 0.5.
Important Questions from Conditional Probability
A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin is tossed, the probability 𝑃(ℎ𝑒𝑎𝑑) = 0.5 and for a fake coin, 𝑃(ℎ𝑒𝑎𝑑) = 1. You pick a coin at random and toss it twice, and get two heads. The probability that the coin you have chosen is the fake coin is _______. (rounded off to two decimal places)
- A clinic specializes in testing for a disease D. The result of the test can be either positive or negative.
A study revealed that if a person suffers from the disease D, the test result in that clinic comes out positive 80% of the time, and negative 20% of the time. If a person is not suffering from the disease D, the test comes out positive 10% of the time and negative 90% of the time. It is also known that among the general population, the disease D occurs in 30% of the individuals.
If a person tests positive for D in that clinic, the probability that he/she actually suffers from the disease D is __________ . (Rounded off to two decimal places) - Suppose an unbiased coin is tossed 6 times. Each coin toss is independent of all previous coin tosses. Let $E_1$ be the event that among the second, fourth, and sixth coin tosses, there are at least two heads. Let $E_2$ be the event that among the first, second, third, and fifth coin tosses, there are equal number of heads and tails.
The conditional probability P($E_1$ | $E_2$) is equal to ____________. (rounded off to one decimal place) - A box contains three apples and two oranges. Two fruits are removed randomly in succession. The probability that the first is an apple and the second is an orange is _____ .
- Bag I contains 4 white and 6 black balls.
Bag II contains 4 white and 3 black balls.
One ball is drawn at random from any one of the two bags and it is found to be a black ball. The probability that the black ball was drawn from Bag I is ______ (rounded off to two decimal places).
