The conditional probability P($E_1$ | $E_2$) is equal to ____________. (rounded off to one decimal place)
To solve the problem, we need to calculate P($E_1$ | $E_2$), the probability of event $E_1$ given that event $E_2$ has occurred. First, let's break down the events $E_1$ and $E_2$:
1. **Event $E_1$:** At least two heads among the 2nd, 4th, and 6th coin tosses. Denote the outcomes of these tosses as $H$, the number of heads among them. We need $H \geq 2$.
2. **Event $E_2$:** Equal number of heads and tails among the 1st, 2nd, 3rd, and 5th tosses. Denote the number of heads as $X$. We require $X = 2$.
**Step 1: Calculate P($E_2$):**
The total combinations of the 1st, 2nd, 3rd, and 5th tosses are $2^4=16$. To have exactly 2 heads, the number of favorable outcomes is $C(4,2)=6$. Thus, P($E_2$) = $\frac{6}{16}=\frac{3}{8}$.
**Step 2: Determine P($E_1$ ∩ $E_2$):**
Given $E_2$, we fixed the 1st, 2nd, 3rd, and 5th tosses with 2 heads and 2 tails. Thus, the outcomes for the 2nd toss (influencing $E_1$) are fixed based on the outcomes of the 1st, 3rd, and 5th tosses. Three scenarios ensure $E_2$ while containing different results for the 2nd toss:
- 2nd toss is Head: leaves 2 choices for the 4th and 6th ($TT$ or $HT$), out of which only $HT$ satisfies $E_1$. Only 1 favorable way exists.
- 2nd toss is Tail: needs exactly 2 heads among 4th and 6th tosses ($HH$) to satisfy $E_1$. 1 favorable way exists here.
Thus, there are 2 favorable outcomes.
**Step 3: Calculate P($E_1$ | $E_2$):**
P($E_1$ | $E_2$) = $\frac{P(E_1 \cap E_2)}{P(E_2)}=\frac{2/16}{3/8}=\frac{2}{6}=\frac{1}{3}$.
Therefore, rounded to one decimal place, P($E_1$ | $E_2$) = 0.5.
Conclusion: The computed value of 0.5 fits within the given range of [0.5,0.5], confirming its correctness.
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) - 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 _____ .
- An experiment is performed by rolling an unbiased die once. Events A and B are defined as follows:
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). - 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).
