1. The chances of winning a single number bet on a casino roulette table is 0.02631 a man bets 10 times what is the probability he wins... Once Twice At least once

2. The Binomial distribution 2 Recap of conditions and probability distribution Textbook work Cumulative distribution Using tables Using a calculator

3. The Binomial distribution The conditions necessary to be modelled by a binomial distribution are: P(X = x) = n C x p x (1 - p) n-x A fixed number of trials ; n Two possible outcomes p for success and 1 - p for failure The probability of each outcome is the same for each trial The trials are independent The probability that there are x successes is given by: Ex 3C page 77 q1 When a random variable X satisfies the conditions for a Binomial distribution then X ~ B(n,p) where n is the number of trials and p is the probability of success.

4. What if n is really large and we want to work out something like this... X~B(20,0.6) what is P(X ≤ 12)? To find this out using our current knowledge would require adding several repetitions of the last formula (either 11 or 9).

5. Ex 3C page 77 1(d) Given that n=11 p=4/5 find p(X≥9) Ans: 0.617

6. Binomial cumulative distribution This is for finding probabilities less than or equal to a specific value: Three methods for working these out 1.Work out each separate value and add them up 2.Use tables (in formula book) 3.Use your calculator

7. Adding and The Use of Tables X~B(20,0.3) what is P(X ≤ 12)? In the formula book are some tables where they are added for you... The rows give P(X≤x) for different values of p.

8. Adding and The Use of Tables X~B(20,0.25) what is P(X > 8)? The rows only give P(X≤x) for different values of p.

9. Adding and The Use of Tables X~B(20,0.6) what is P(X ≤ 12)? Oh no the table only reads up to p=0.5 so we must model less than or equal to 12 successes at p=0.6 8 or more successes at p=0.4

10. Using tables to find binomial probabilities

11. Visualising – Tables only do ≤ X~B(20,0.6) What is P(X 12)? What is P(3<X≤9)

12. Activities: Tarsia puzzle and/or Domino trail

13. Calculator The calculator is far easier to use than the tables. It works for any probability but still only does ≤ X~B(20,0.6) What is P(X 12)? What is P(3<X≤9)

14. Using a calculator to find binomial probabilities MenuBinm (F5) Bpd (F1) is for finding individual binomial probabilities Bcd (F2) is for finding cumulative binomial probabilities InvB (F3) is for finding the inverse of binomial StatDist (F5) Example using Bpd Data : variable x : 1 Numtrial : 9 P : 0.2 Move down to Execute and then press CALC (F1) p =0.30198988 (as part (d)) Example using Bcd Data : variable x : 4 Numtrial : 9 P : 0.2 Move down to Execute and then press CALC (F1) p = 0.98041856 (as part (a))

16. Using a calculator to find binomial probabilities MenuBinm (F5) Bpd (F1) is for finding individual binomial probabilities Bcd (F2) is for finding cumulative binomial probabilities InvB (F3) is for finding the inverse of binomial StatDist (F5) Example using Bpd Data : variable x : 1 Numtrial : 9 P : 0.2 Move down to Execute and then press CALC (F1) p =0.30198988 (as part (d)) Example using Bcd Data : variable x : 4 Numtrial : 9 P : 0.2 Move down to Execute and then press CALC (F1) p = 0.98041856 (as part (a))

17. Activity: Relay race

28. Exam style questions