For Problems 7–9: A JEE aspirant estimates that she will be successful with an 80% chance if she studies 10 hours per day, with a 60% chance if she studies 7 hours per day, and with a 40% chance if she studies 4 hours per day. She further believes that she will study 10 hours, 7 hours, and 4 hours per day with probabilities 0.1, 0.2, and 0.7, respectively.Given that she is successful, the chance that she studied for 4 hours is
In throwing a dice thrice, getting numbers in order denoted by \(a, b, c\), satisfying \(a^2 + 4b^2 + 4c^2 - 2ab - 4bc - 2ac = 0\). If probability such that point \((a, b, c)\) lies inside the tetrahedron formed by the plane \(x + y + z = 10\) and co-ordinate planes is \(\dfrac{6}{\lambda}\), where \(\lambda \in N\), then \(\lambda\) is:
For Problems 13–15: An amoeba either splits into two or remains the same or eventually dies out immediately after completion of every second with probabilities, respectively, 1/2, 1/4, and 1/4. Let the initial amoeba be called as mother amoeba and after every second, the amoeba, if it is distinct from the previous one, be called as 2nd, 3rd, ... generations.The probability that immediately after completion of 2 s all the amoeba population dies out is
\(\displaystyle e^{\lim_{x \to 0} 2\left(\dfrac{a^x - 1 + b^x - 1}{2}\right)} = e^{\ln ab} = ab = 6\). The value of \(P(E)\) where ordered pairs \((a,b)\) with \(ab = 6\) are selected from \(\{(1,6),(6,1),(2,3),(3,2)\}\) is:
Let p, q be chosen one by one from the set \(\{1, \sqrt{2}, \sqrt{3}, 2, e, \pi\}\) with replacement. Now a circle is drawn taking (p, q) as its centre. Then the probability that at the most two rational points exist on the circle is (rational points are those points whose both the coordinates are rational)
An unbiased dice, with faces numbered 1, 2, 3, 4, 5, 6, is thrown \(n\) times and the list of \(n\) numbers shown up is noted. Then find the probability that among the numbers 1, 2, 3, 4, 5, 6 only three numbers appear in this list and each number appears at least once.