• 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

The Binomial distribution 2

Recap of conditions and probability distributionTextbook work

Cumulative distributionUsing tables

Using a calculator

The Binomial distributionThe conditions necessary to be modelled by a binomial distribution are:

P(X = x) = nCx px (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 thenX ~ B(n,p) where n is the number of trials and p is the probability of success.

• 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 13 or 8).

Binomial cumulative distribution

This is for finding probabilities less than or equal to a specific value:Three methods for working these out1. Work out each separate value and add them up2. Use tables (in formula book)3. Use your calculator

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.

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.

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

Using tables to find binomial probabilities

Visualising – Tables only do ≤

• X~B(20,0.6)• What is P(X<12)? What is P(X>12)?• What is P(3<X≤9)

Activities:

Tarsia puzzle

and/or

Domino trail

CalculatorThe 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(X>12)?What is P(3<X≤9)

Using a calculator to find binomial probabilities

Menu Binm (F5)

Bpd (F1) is for finding individual binomial probabilitiesBcd (F2) is for finding cumulative binomial probabilitiesInvB (F3) is for finding the inverse of binomial

Stat Dist (F5)

Example using BpdData : variablex : 1Numtrial : 9P : 0.2

Move down to Execute and then press CALC (F1)

p =0.30198988 (as part (d))

Example using BcdData : variablex : 4Numtrial : 9P : 0.2Move down to Execute and then press CALC (F1)

p = 0.98041856 (as part (a))

Activity:

Relay race

Exam style questions