Question:
A coin is tossed three times. If $X$ denotes the absolute difference between the number of heads and the number of tails, then $P(X = 1) =$
A coin is tossed three times. If $X$ denotes the absolute difference between the number of heads and the number of tails, then $P(X = 1) =$
Show Hint
With 3 tosses, the only possible values for the absolute difference $X$ are 3 (all same faces) or 1 (mixed faces). The outcomes for $X = 3$ are simply $\{HHH, TTT\}$, which accounts for exactly 2 outcomes out of 8. Therefore, the remaining outcomes must yield $X = 1$. This lets you calculate the probability directly as: $P(X=1) = 1 - P(X=3) = 1 - \frac{2}{8} = \frac{6}{8} = \frac{3}{4}$.
Updated On: Jun 18, 2026
- $\frac{1}{6}$
- $\frac{1}{2}$
- $\frac{2}{3}$
- $\frac{3}{4}$
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
A fair coin is flipped 3 times, leading to a sample space of 8 equally likely outcomes. The random variable $X$ is defined as the absolute mathematical difference between the total counts of Heads ($H$) and Tails ($T$), i.e., $X = |N_H - N_T|$. We need to find the probability that $X = 1$.
Step 2: Key Formula or Approach:
1. List out the sample space $S$ for 3 coin tosses. 2. For each sample point, evaluate the absolute difference $X = |N_H - N_T|$. 3. Use the classical probability definition: $P(X = 1) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}}$.
Step 3: Detailed Explanation:
The total sample space $S$ contains $2^3 = 8$ elements: $$S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$$ Let's calculate $X$ for each outcome: For $HHH$: $3$ heads, $0$ tails $\implies X = |3 - 0| = 3$ For $HHT$: $2$ heads, $1$ tail $\implies X = |2 - 1| = 1$ For $HTH$: $2$ heads, $1$ tail $\implies X = |2 - 1| = 1$ For $THH$: $2$ heads, $1$ tail $\implies X = |2 - 1| = 1$ For $HTT$: $1$ head, $2$ tails $\implies X = |1 - 2| = 1$ For $THT$: $1$ head, $2$ tails $\implies X = |1 - 2| = 1$ For $TTH$: $1$ head, $2$ tails $\implies X = |1 - 2| = 1$ For $TTT$: $0$ heads, $3$ tails $\implies X = |0 - 3| = 3$ The outcomes where $X = 1$ are $\{HHT, HTH, THH, HTT, THT, TTH\}$. The number of favorable outcomes is 6. $$P(X = 1) = \frac{6}{8} = \frac{3}{4}$$
Step 4: Final Answer:
The probability that $P(X = 1)$ is $\frac{3}{4}$, which corresponds to option (D).
A fair coin is flipped 3 times, leading to a sample space of 8 equally likely outcomes. The random variable $X$ is defined as the absolute mathematical difference between the total counts of Heads ($H$) and Tails ($T$), i.e., $X = |N_H - N_T|$. We need to find the probability that $X = 1$.
Step 2: Key Formula or Approach:
1. List out the sample space $S$ for 3 coin tosses. 2. For each sample point, evaluate the absolute difference $X = |N_H - N_T|$. 3. Use the classical probability definition: $P(X = 1) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}}$.
Step 3: Detailed Explanation:
The total sample space $S$ contains $2^3 = 8$ elements: $$S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$$ Let's calculate $X$ for each outcome: For $HHH$: $3$ heads, $0$ tails $\implies X = |3 - 0| = 3$ For $HHT$: $2$ heads, $1$ tail $\implies X = |2 - 1| = 1$ For $HTH$: $2$ heads, $1$ tail $\implies X = |2 - 1| = 1$ For $THH$: $2$ heads, $1$ tail $\implies X = |2 - 1| = 1$ For $HTT$: $1$ head, $2$ tails $\implies X = |1 - 2| = 1$ For $THT$: $1$ head, $2$ tails $\implies X = |1 - 2| = 1$ For $TTH$: $1$ head, $2$ tails $\implies X = |1 - 2| = 1$ For $TTT$: $0$ heads, $3$ tails $\implies X = |0 - 3| = 3$ The outcomes where $X = 1$ are $\{HHT, HTH, THH, HTT, THT, TTH\}$. The number of favorable outcomes is 6. $$P(X = 1) = \frac{6}{8} = \frac{3}{4}$$
Step 4: Final Answer:
The probability that $P(X = 1)$ is $\frac{3}{4}$, which corresponds to option (D).
Was this answer helpful?
0
0
Top MHT CET Mathematics Questions
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
- If $\int\limits^{K}_0 \frac{dx}{2 + 18 x^2} = \frac{\pi}{24}$, then the value of K is
- If $\int^{\pi/2}_{0} \log\cos x dx =\frac{\pi}{2} \log\left(\frac{1}{2}\right)$ then $ \int^{\pi/2}_{0} \log\sec x dx = $
- The point on the curve $y = \sqrt{x - 1}$ where the tangent is perpendicular to the line $2x + y - 5 = 0 $ is
View More Questions
Top MHT CET Probability Questions
- If $r. v. x :$ waiting time in minutes for bus and $p.d.f.$ of $x$ is given by $f(x) = \begin{cases} \frac{1}{5} , & 0\le x\le5 \\[2ex] 0, & \text{otherwise} \end{cases}$ then probability of waiting time not more than $4$ minutes is = _______
- A die is thrown twice. If getting a number greater than four on the die is considered a success, then the variance of the probability distribution of the number of successes is
- If $r.v. X \sim B \left(n = 5, P=\frac{1}{3}\right)$ then $P(2 < X < 4) =$ ______________
- Given $ P(A \cup B ) = 0.6,P(A\cap B) = 0.2 $ , the probability of exactly one of the event occurs is
- From a group of $ 8 $ boys and $ 3 $ girls, a commitee of $ 5 $ members to be formed. Find the probability that $ 2 $ particular girls are included in the committe is
View More Questions
Top MHT CET Questions
- During r- DNA technology, which one of the following enzymes is used for cleaving DNA molecule ?
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- The temperature of $32^{\circ}C$ is equivalent to
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- The heat of formation of water is $ 260\, kJ $ . How much $ H_2O $ is decomposed by $ 130\, kJ $ of heat ?
View More Questions