List of practice Questions

From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to $a/b$, where $a, b \in \mathbb{N}$ and $\gcd(a, b) = 1$, then $a + b$ is equal to _______

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is \(p\), then \(96p\) is equal to ____.

A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5 points for each tail he throws. If the probability that he gets exactly 30 points is \( \frac{m}{n} \), gcd \( (m, n) = 1 \), then \( m + n \) is equal to:

Let \(a,b,c \in \{1,2,3,4\}\). If the probability that \[ ax^2 + 2\sqrt{2}\,bx + c>0 \quad \text{for all } x \in \mathbb{R} \] is \( \frac{m}{n} \), where \(\gcd(m,n)=1\), then \(m+n\) is equal to _____.

Let E, F and G be mutually independent events such that $P(E) = 0.4$, $P(F) = 0.6$ and $P(G) = 0.8$ then $P(\overline{E} \cup \overline{F} \cup G)$ is

Let E and F be two events, if $P(E|F)=0.5$, $P(E|\overline{F})=0.6$ and $P(F)=0.6$ then $P(E)$ equals

A family has two children. If it is known that at least one child is a boy, then find the probability of both children being boys.

If \(P(A)=\dfrac{7}{13}\), \(P(B)=\dfrac{9}{13}\) and \(P(A\cap B)=\dfrac{4}{13}\), then find \(P(A/B)\).

It is given that the numbers obtained on throwing two dice are different. Find the probability that the sum of both numbers is 4.

Prove that if \(E\) and \(F\) are independent events, then \(E\) and \(F'\) will also be independent.

If \(P(A)=0.8,\ P(B)=0.5\) and \(P(B\mid A)=0.4\), find \(P(A\cup B)\).

If a die is thrown three times, then find the probability of getting at least one odd number.

If \(P(A)=\dfrac13\), \(P(B)=\dfrac12\) and \(P(A\cup B)=\dfrac23\), then show that \(A\) and \(B\) are independent events.

If \(P(A)=\dfrac7{13}\), \(P(B)=\dfrac9{13}\) and \(P(A\cap B)=\dfrac4{13}\), then find \(P\left(\dfrac AB\right)\).

A die is thrown twice and found that the sum of the numbers is 6. Find the conditional probability of getting number 4 at least once.

If \(P(A)=\dfrac12\), \(P(B)=\dfrac13\) and \(A,B\) are independent events, find the value of \(P(A/B)\).

Prove that if \(E\) and \(F\) are two independent events, then \(E\) and \(F'\) will also be independent.

If \(A\) and \(B\) are independent events and \(P(A)=0.3\), \(P(B)=0.4\), then find (i) \(P(A\cap B)\), (ii) \(P(A\cup B)\).

The probability of impossible events is:

Prove that if \(A\) and \(B\) are independent events, then the probability of happening of at least one of \(A\) or \(B\) is \(\left[1-P(A')P(B')\right]\).

If \(P(A)=\dfrac{7}{13}\), \(P(B)=\dfrac{9}{13}\) and \(P(A\cap B)=\dfrac{4}{13}\), then find \(P(A\cup B)\) and \(P(A/B)\).

If \(3P(A)=P(B)=\dfrac{5}{13}\) and \(P(A/B)=\dfrac{2}{5}\), then \(P(A\cup B)\) will be:

Let \(X\) and \(Y\) be two independent discrete random variables such that the moment generating functions of \(X\) and \(X+Y\) are given by \[ M_X(t)=\frac{1+2e^{-t}+3e^{2t}}{6},\quad t\in\mathbb{R}, \] and \[ M_{X+Y}(t)=\frac{2+e^{t}+3e^{3t}}{6},\quad t\in\mathbb{R}, \] respectively. Then which of the following statements is correct?

Let \(X\) and \(Y\) be identically distributed random variables with variance \(\sigma^2\in(0,\infty)\). Then the correlation coefficient between \(X\) and \(Y\) is

Let \(X\) and \(Y\) be independent and identically distributed normal random variables. If
\[ P(X+2Y\le3)=P(2X-Y\ge4), \]
then \(E(X)\) is