independent events

A day can only be cloudy or sunny. The probability of a day being cloudy is 0.5 , independent of the condition on other days. What is the probability that in any given four days, t...

Suppose an unbiased coin is tossed 6 times. Each coin toss is independent of all previous coin tosses. Let $E_1$ be the event that among the second, fourth, and sixth coin tosses,...

Suppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is 0.01. F...

Consider a permutation sampled uniformly at random from the set of all permutations of {1, 2, 3, ..., n} for some n ≥ 4. Let X be the event that 1 occurs before 2 in the permutatio...

Consider a permutation sampled uniformly at random from the set of all permutations of {1, 2, 3, ..., n } for some n ≥ 4. Let X be the event that 1 occurs before 2 in the permutati...

Consider a random experiment where two fair coins are tossed. Let A be the event that denotes HEAD on both the throws, B be the event that denotes HEAD on the first throw, and C be...

Two numbers are chosen independently and uniformly at random from the set {1, 2, ...., 13}. The probability (rounded off to 3 decimal places) that their 4-bit (unsigned) binary rep...

The probability that a $k$-digit number does NOT contain the digits 0, 5, or 9 is :

The probabilities that a student passes in Mathematics, Physics and Chemistry are $$m, p$$ and $$c$$ respectively. Of these subjects, the student has $$75$$% chance of passing in a...

A random bit string of length n is constructed by tossing a fair coin n times and setting a bit to 0 or 1 depending on outcomes head and tail, respectively. The probability that tw...

$${{E_1}}$$ and $${{E_2}}$$ are events in a probability space satisfying the following constraints: $$ \bullet $$ $$\Pr \,\,({E_1}) = \Pr \,({E_2})$$ $$ \bullet $$ $$\Pr \,\,({E_1}...

Consider two events $${{E_1}}$$ and $${{E_2}}$$ such that probability of $${{E_1}}$$, Pr [$${{E_1}}$$] = 1/2, probability of $${{E_2}}$$, Pr[$${{E_2}}$$ = 1/3, and probability of $...