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 amoeba population will be maximum after completion of 3 s is
Example 30 (Assertion-Reason):A fair die is thrown twice. Let $(a, b)$ denote the outcome in which the first throw shows $a$ and the second shows $b$. Let A and B be the following two events:$A = \{(a,b) \mid a \text{ is even}\}$, $B = \{(a,b) \mid b \text{ is even}\}$Statement-1: If $C = \{(a,b) \mid a + b \text{ is odd}\}$, then $P(A \cap B \cap C) = \frac{1}{8}$.Statement-2: If $D = \{(a,b) \mid a + b \text{ is even}\}$, then $P[(A \cap B \cap D) \mid (A \cup B)] = 1$.
The decimal parts of the logarithms of two numbers taken at random are found to six places of decimal. What is the chance that the second can be subtracted from the first without 'borrowing'?For each column of the two numbers, n(S) = number of ways to fill the two places by the digits 0, 1, 2, ..., 9 = 10 × 10 = 100.
An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the sum of the numbers obtained is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered 2, 3, 4, ..., 12 is picked and the number on the card is noted. The probability that the noted number is either 7 or 8 is
There are \((n+1)\) urns, each containing some balls. Let \(E_1\) denote the event that one of the first \(n\) urns is chosen and \(E_2\) denote the event that the \((n+1)\)th urn is selected. \(A\) denotes the event that two balls drawn are black. Then \(p(E_1) = \dfrac{n}{n+1}\), \(P(E_2) = \dfrac{1}{n+1}\), \(P(A/E_1) = \dfrac{^6C_2}{^{10}C_2} = \dfrac{1}{3}\) and \(P(A/E_2) = \dfrac{^5C_2}{^{10}C_2} = \dfrac{2}{9}\). Using Bayes' Theorem, if \(P(E_2/A) = \dfrac{1}{16}\), find the value of \(n\).
A doctor is called to see a sick child. The doctor knows (prior to the visit) that 90% of the sick children in that neighborhood are sick with the flu, denoted by \(F\), while 10% are sick with the measles, denoted by \(M\). A well-known symptom of measles is a rash, denoted by \(R\). The probability of having a rash for a child sick with the measles is 0.95. However, occasionally children with the flu also develop a rash, with conditional probability 0.08. Upon examination the child, the doctor finds a rash. Then what is the probability that the child has the measles?