\( \frac{9}{16} \)
We are given that a fair coin is tossed 6 times, and we need to find the conditional probability that exactly 3 heads appear given that the first three tosses resulted in at least 2 heads.
Step 1: Define total probability space
Each toss of the coin is independent with two outcomes: head (H) or tail (T). The probability of any specific sequence of 6 tosses occurring is: \[ \left(\frac{1}{2}\right)^6 = \frac{1}{64} \]
Step 2: Compute probability of the given condition
The probability that the first three tosses result in at least 2 heads can be computed by considering the cases: 1. \( (H, H, H) \) 2. \( (H, H, T) \) 3. \( (H, T, H) \) 4. \( (T, H, H) \) Each of these cases follows a binomial probability distribution: \[ P(\text{At least 2 heads in first 3 tosses}) = P(2H) + P(3H) \] Using the binomial formula: \[ P(2H) = \binom{3}{2} \left(\frac{1}{2}\right)^3 = 3 \times \frac{1}{8} = \frac{3}{8} \] \[ P(3H) = \binom{3}{3} \left(\frac{1}{2}\right)^3 = 1 \times \frac{1}{8} = \frac{1}{8} \] \[ P(\text{At least 2 heads in first 3 tosses}) = \frac{3}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \]
Step 3: Compute probability of exactly 3 heads given the condition
We need to find \( P(X = 3 | \text{At least 2 heads in first 3 tosses}) \), which is given by: \[ P(X = 3 \cap A) / P(A) \] where \( A \) is the event that at least 2 heads occur in the first 3 tosses. For exactly 3 heads in 6 tosses, given that at least 2 heads occurred in the first 3 tosses, the remaining 3 tosses must contribute either 0 or 1 additional head. This follows the binomial probability: \[ P(3H \text{ in 6 tosses} | A) = \frac{5}{16} \]
Step 4: Conclusion
Thus, the correct answer is: \[ \mathbf{\frac{5}{16}} \]
If three numbers are randomly selected from the set \( \{1,2,3,\dots,50\} \), then the probability that they are in arithmetic progression is:
A student has to write the words ABILITY, PROBABILITY, FACILITY, MOBILITY. He wrote one word and erased all the letters in it except two consecutive letters. If 'LI' is left after erasing then the probability that the boy wrote the word PROBABILITY is: \
Given the vectors:
\[ \mathbf{a} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} \]
\[ \mathbf{b} = 3(\mathbf{i} - \mathbf{j} + \mathbf{k}) = 3\mathbf{i} - 3\mathbf{j} + 3\mathbf{k} \]
where
\[ \mathbf{a} \times \mathbf{c} = \mathbf{b} \]
\[ \mathbf{a} \cdot \mathbf{x} = 3 \]
Find:
\[ \mathbf{a} \cdot (\mathbf{x} \times \mathbf{b} - \mathbf{c}) \]
A rectangle is formed by the lines \[ x = 4, \quad x = -2, \quad y = 5, \quad y = -2 \] and a circle is drawn through the vertices of this rectangle. The pole of the line \[ y + 2 = 0 \] with respect to this circle is:
The equation of a circle which passes through the points of intersection of the circles \[ 2x^2 + 2y^2 - 2x + 6y - 3 = 0, \quad x^2 + y^2 + 4x + 2y + 1 = 0 \] and whose centre lies on the common chord of these circles is: