A Bag Contains 6 Red Balls, 4 Blue Balls, and 2 White Balls

EDEXCEL STATISTICS 1 PROBABILITY

Combined events involving simple permutations and combinations

Consider the usual “balls in a bag” type example to introduce an elementary idea of permutations and combinations.

A bag contains 6 red balls, 4 blue balls, and 2 white balls

(a)If three balls are drawn at random from the bag, find the probability that two are red and one is white

(b)If three balls are drawn at random from the bag, find the probability that two are red.

(c)If four balls are drawn at random from the bag, find the probability that two are red.

Although strictly speaking the events are conditional, it’s not worth getting bogged down with the conditional notation for this type of question, simply adopt a common sense approach.

(a)

However, the question should be interpreted as “find the probability that two are red and one is white in any order”

This phrase is essential and common sense tells us that the white ball could be drawn first second or third. (WRR, RWR, RRW) in other words there are 3 possible arrangements. So more correctly

The next part is similar we simply specify “not red” instead of white

(b)

(c)This is also similar we require

However, the number of arrangements is a little less obvious. Again a simple list should convince you that there are six possible arrangements.

Hence we have

In parts (a) and (b) we can rely on common sense to spot the number of arrangements in part (c) we can make life easier by considering combinations. This is a topic that may be familiar from work done in C2, depending on what order you are studying your modules.

The number of combinations in part (c) is the same as saying that there a four students in a group and any two are picked to complete a task, how many pairs could you pick ?

We would write

In generalWhere n! =n.(n-1).(n-2)…..2.1

This topic is developed much further in C2 for S1 only simple combinations will be required and a nice easy way to find them is a quick sketch of Pascal’s triangle

Here are a couple of fairly standard type questions which use elementary combinations.

Four standard dice are rolled and the number of sixes recorded. What is the probability of recording

(a)Exactly one six

(b)Exactly two sixes

(c)Two sixes given that there were no sixes on the first two throws.

(a)

(b)

(c)

Alex has 5 Hip Hop CDs, 8 Rap & 7 Garage lying in a drawer. If he selects 3 CDs at once what is the probability of him picking

(a)All 3 Hip hop

(b)2 Hip hop & a Garage

(c)One of each style.

Here are the solutions using the obvious notation, the key points are that the CDs are selected without replacement, and also the key phrase “in any order”

(a)

(b)

(c)

Titus Salt School - A Teachnet Uk 2008 Project

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