Probability Tes...

For numbers are chosen at random (without replacement) from the set {1, 2, 3, …,20}
Statement 1: The probability that the chosen numbers when arranged in some order will form an AP is 1/85
Statement 2: If the four chosen numbers form an AP, then the set of all possible values of common difference is {±1, ±2, ±3, ±4, ±5}

Consider the system of equations
ax + by = 0, cx + dy = 0, where a, b, c, d ϵ{0, 1}
Statement 1: The probability that the system of equations has a unique solution is 3/8.
Statement 2: The probability that the system of equations has a unique solution is 1

Let U₁ and U₂ be two urns such that U₁ contains 3 white and 2 red balls and U₂ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from U₁ and put into U₂. However, if tail appears then 2 balls are drawn at random from U₁ and put into U₂. Now, 1 ball is drawn at random from U₂.
The probability of the drawn ball from U₂ being white is

Let U₁ and U₂ be two urns such that U₁ contains 3 white and 2 red balls and U₂ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from U₁ and put into U₂. However, if tail appears then 2 balls are drawn at random from U₁ and put into U₂. Now, 1 ball is drawn at random from U₂.
Given that the drawn ball from U₂ is white, the probability that head appeared on the coin is

A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required.
The probability that X = 3 equals

A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required.
The probability that X ≥ 3 equals

A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required.
The conditional probability that X ≥ 6 given X > 3 equals

There are n urns each containing (n + 1) balls such that the i^th urn contains ‘i’ white balls and (n + 1 – i) red balls. Let u_i be the event of selecting i^th urn, I = 1, 2, 3,…,n and W denotes the event of getting a white balls.

There are n urns each containing (n + 1) balls such that the i^th urn contains ‘i’ white balls and (n + 1 – i) red balls. Let u_i be the event of selecting i^th urn, I = 1, 2, 3,…,n and W denotes the event of getting a white balls.