Binomial probability formula
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1) The document introduces the binomial probability formula used to calculate the probability of getting doubles when rolling a pair of dice multiple times. 2) It provides the formula: P(x) = (nCx)p^x(1-p)^(n-x) where n is the number of trials, p is the probability of success on each trial, and x is the number of successes. 3) An example calculates the probability of getting doubles 1 time out of 5 rolls as 0.401878 using the formula with n=5, p=6/36, and x=1.
Binomial probability formula
- 1. Introduction If I were to roll a pair of dice 5 times… What’s the probability I get doubles… 1 time? 2 times? 3 times? 4 times? 5 times? To answer these questions, We need this really cool thing The Binomial Probability Formula!
- 2. Formula x n x n xP(x) C p q Probability of x n number of trials, choose x Probability of success raised to the x power Probabilty of failure raised to the n-x power
- 3. Formula Since q = 1-p, the formula can be rewritten as: x n x n xP(x) C p (1 p)
- 4. Formula You need to input 3 numbers into this formula to find the probability of x number of x successes out of n trials: ► the number of trials n ► the probability of success, p ► x, the number of successes x n x n xP(x) C p (1 p)
- 5. Remember The nature of a binomial experiment is to have a specified number of trials The probability of success, p, is the same for every trial x is the random variable and usually the focus of the problem x = {1, 2, 3, …, n} If n = 5, then x may equal 0, 1, 2, 3, 4, or 5
- 6. Insight In the binomial probability formula, nCx determines the number of ways of getting x successes in n trials, regardless of order.
- 7. Example n=5, p = 6/36, x=1 P(1) = (5C1)(6/36)1(30/36)5-1 P(1) = (5)(1/6)1(5/6)4 P(1) = (5/6)(5/6)4 P(1) = (5/6)5 = .401878 If I were to roll a pair of dice 5 times… What’s the probability I get doubles… 1 time? x n x n xP(x) C p (1 p)
- 8. CalculationTips n=5, p = 6/36, x=1 P(1) = (5C1)(6/36)1(30/36)5-1 P(1) = (5)(1/6)1(5/6)4 P(1) = (5/6)(5/6)4 P(1) = (5/6)5 = .401878 x n x n xP(x) C p (1 p) Think of the formula as multiplying 3 numbers Group (n-x) Resolve combination formula and exponents before multiplying! Since there are 36 possible outcomes when rolling two, 6- sided die, and 6 of those outcomes are doubles, then the probability of success is 6/36
- 9. Example n=5, p = 6/36, x=3 P(1) = (5C3)(6/36)3(30/36)5-3 P(1) = (10)(1/6)3(5/6)2 P(1) = (10)(1/216)(25/36) P(1) = (10/216)(25/36) = .03125 If I were to roll a pair of dice 5 times… What’s the probability I get doubles… 3 times? x n x n xP(x) C p (1 p)
- 10. Thinkabout If we were to find P(0), P(1), P(2), P(3), P(4), and P(5), what would the sum of those probabilities be?