5 questions from 2021
Let \( E, F \) and \( G \) be three events having probabilities
\( P(E)=\dfrac{1}{8} \), \( P(F)=\dfrac{1}{6} \) and \( P(G)=\dfrac{1}{4} \) and let \( P(E \cap F \cap G) = \dfrac{1}{10} \).
For any event \( H \), \( H^{c} \) denotes its complement, then which of the following statements is(are) TRUE?
Consider three sets \( E_1=\{1,2,3\} \), \( F_1=\{1,3,4\} \) and \( G_1=\{2,3,4,5\} \). Two elements are chosen at random, without replacement, from the set \( E_1 \) and let \( S_1 \) denote the set of these chosen elements. Let \( E_2=E_1-S_1 \) and \( F_2=F_1 \cup S_1 \). Now two elements are chosen at random, without replacement, from the set \( F_2 \) and let \( S_2 \) denote the set of these chosen elements.
Let \( G_2=G_1 \cup S_2 \). Finally, two elements are chosen at random, without replacement, from the set \( G_2 \) and let \( S_3 \) denote the set of these chosen elements.
Let \( E_3=E_2 \cup S_3 \). Given that \( E_1=E_3 \), let \( p \) be the conditional probability of the event \( S_1=\{1,2\} \). Then the value of \( p \) is:
Three numbers are chosen at random, one after another with replacement, from the set \( S=\{1,2,3,\ldots,100\} \). Let \( p_1 \) be the probability that the maximum of chosen numbers is at least \( 81 \) and \( p_2 \) be the probability that the minimum of chosen numbers is at most \( 40 \).
The value of \( \frac{625}{4}\, p_1 \) is ______.
A number is chosen at random from the set \( \{1, 2, 3, \ldots, 2000\} \). Let \( p \) be the probability that the chosen number is a multiple of \( 3 \) or a multiple of \( 7 \). Then the value of \( 500p \) is \( \_\_\_\_\_\_\_\_ \).
Three numbers are chosen at random, one after another with replacement, from the set \( S = \{ 1, 2, 3, \ldots, 100 \} \). Let \( p_1 \) be the probability that the maximum of chosen numbers is at least \( 81 \) and \( p_2 \) be the probability that the minimum of chosen numbers is at most \( 40 \).
The value of \( \frac{125}{4} \, p_2 \) is _______.
