If we let X be the random variable of the number of trials up to and including the first success, then X has a Geometric Distribution. For example: If you were to flip a coin wanting to get a head, you would keep flipping until you obtained that head. Or if you needed a double top in darts, you would keep throwing until you hit it. The probabilities are worked out like this... Remember: p = probability of success q = probability of failure P(X = 1) = p No failure, success on first attempt P(X = 2) = q × p 1 failure then success P(X = 3) = q2 × p 2 failures then success P(X = 4) = q3 × p 3 failures then success P(X = r) = qr-1 × p r - 1 failures followed by success If X follows a Geometric distribution, we write: X ~ Geo(p)This reads as 'X has a geometric distribution with probability of success, p'. Example: In a particular game you may only begin if you roll a double to start. Find the probability that: you start on your first go; you need 4 attempts before you start; you start within 3 attempts; you need greater than 6 attempts before starting. Before we start, let's write down the distribution of X with the probability of a success - in this case - being the probability of rolling a double with 2 dice. X ~ Geo (1/6) as probability of double = 1/6 1. 'starting on your first go' requires rolling a double on your first attempt. P(X = 1) = 1/6 2. 'you need 4 attempts to start' requires you to fail for the first 3 attempts./**/ 3. 'starting within 3 goes', means you could start on your first, second or third goes4. 'you need greater than 6 attempts' is best thought of as being the same as needing 6 failures. The last part of this example gives us a special result to remember: P(X > r) = qrExpectation and varianceIf: X ~ Geo(p)Then: Example: If the probability to pot a ball off the break in pool is 0.4, find the expected number of breaks before success and the corresponding variance and standard deviation. Here: X ~ Geo(0.4) Therefore: E(X) = 1/0.4 = 2.5So we would expect to pot off the break every 2.5 goes. Hence:σ = 1.94 (3 sf)