Starter Toss the two coins and record the results two heads a head and a tail two tails P(two heads) P(a head and a tail) P(two tails)

Probability distributions To understand what a random variable is Know the properties of a random variable, Be able to construct a probability distribution for a random variable. Introduction to the Binomial distribution

Random variables A random variable is a quantity whose value depends on chance. Example Two coins are thrown together. The random variable X is the number of heads obtained. The table shows the probability distribution for X x P(X = x)

Horse Race

One for you to try Two dice are rolled together. The random variable X is the number of 6’s obtained. Tabulate the probability distribution

An important property of a probability distribution ∑P(X = x) = 1 Example: the following table give the probability distribution for the random variable S s P(s = S) c 2c 3c 2c c Find (a) the value of c (b) P(S < 3) (c) P(S ≤ 3) (d) P(0 < S ≤ 4)

Aims: To know what the binomial distribution is and use it to find probabilities. Outcomes: Name: Know what the Binomial Distribution is. Describe: When the Binomial Distribution can be used. Explain: The formula for Binomial distribution and how it is used. Skills: Find probabilities for the Binomial on a GDC

A fair 6 sided die has a probability of 1 / 3 of rolling a factor of 9. Suppose the die is rolled 5 times... What is the probability I roll a factor of 9 twice?

Suppose we get a factor of 9 on the first two goes then not on the next 3 the probability is 1 / 3 x 1 / 3 x 2 / 3 x 2 / 3 x 2 / 3 = 8 / 243 Success on 1 st and 3 rd goes 1 / 3 x 2 / 3 x 1 / 3 x 2 / 3 x 2 / 3 = 8 / 243 … all orders have the same probability… so we multiply 8 / 243 by the number of possible orders.

SSFFF SFSFF SFFSF SFFFS FSSFF FSFSF FSFFS FFSSF FFSFS FFFSS 10 orders So probability of 2 successes is… 10 x 8 / 243 = 80 / 243

All orders of x successes have the same probability. A formula where n is the number of trials and p is the probability of success is (p) x (1-p) n-x Now we need to multiply by the possible orders…

To find the orders we use Pascal's triangle or the combinations function. n C x or The probability is therefore given by…

A dice is biased so that the probability of rolling a 6 is 0.7 the dice is rolled 8 times. X represents the number of times 6 is rolled. What is... P(X=3) P(X = 4 or 5) P(3<X≤5) P(X < 7)

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: Eg given that n = 10, p = ¼, find P(X = 3) Ex 3C page 77 q1

Ex 3C page 77 or relay race with some more challenging questions

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 = Example using Bcd Data : variable x : 4 Numtrial : 9 P : 0.2 Move down to Execute and then press CALC (F1) p =

Using the formula or calculator find the answer to… If I roll a fair dice 18 times what is the probability I roll… The number 3 four times

Using the formula or calculator find the answer to… The probability I win in a game of bingo is 0.05 If I play 15 games what is the chance I win more than once? P(0)= P(1)= P(0)-P(1) =

The probability that Joan gets an A in her exams is 0.3 if she takes 6 exams what is the probability she gets 3 or 4 As? P(3)= P(4)= P(3)+P(4) =

If X~B(10, 0.6) what is… P(X=7) P(1<X<4) = P(X>0) =0.9999

To denote that a random variable X follows a Binomial we need to pass along the important information number of trials (n) and probability (p) We write this… X~B(n,p)

Name: Know what the Binomial Distribution is. Describe: When the Binomial Distribution can be used. Explain: The formula for Binomial distribution and how it is used. Skills: Find probabilities for the Binomial on a GDC