## Overview Probability

There is some jargon to learn: mutually exclusive events, combined events, expected frequency, relative frequency, experimental probability, estimated probability, independent events and theoretical probability. This may seem a little intimidating but with a little practice their meaning becomes clear and easy to remember.

Just to irritate me it turns out that relative frequency, experimental probability and estimated probability all the mean the same thing!

Tree diagrams are deemed to be the most advanced sub-topic (Grade A and A*) within the Maths GCSE. I’m not sure why this is the case and if you can understand the basics of probability its easy to use tree diagrams.

As usual its best to get stuck into some questions.

### Mutually Exclusive Events, Expectation, Relative Frequency, Independent Events

#### Question

- a) A school sells 4,800 raffle tickets. There are four different colours; (T)aupe, (S)almon, (B)eige and ℗ink.

**On the first draw:**

P(T) 1/3 P(S) =5 ⁄12 P(B) = 1/4

Work out the probability of picking a Pink ticket on the first draw.

**How many tickets of each colour have been sold?**

- b) The probability of any customer entering a supermarket, buying a loaf of bread is 0.45. On Saturday there are are expected to be 3,200 customers, how many customers can be expected to buy a loaf of bread that day?
- c) A roulette wheel has 37 slots, numbers 1 to 36 plus an extra slot “0”. A gambler suspects that the wheel is biased towards the “0” slot. He decides to records outcomes:

**Complete the table. Do you believe the wheel is biased towards the “0” slot? Explain your answer.**

- d) Tom and John eat a variety of breakfast cereals. On any day there is a 40% chance that Tom will choose Cornflakes and a 50% chance that he will choose Shredded Wheat. There is an 80% chance that John will choose Cornflakes and a 10% chance he will choose Shredded Wheat. On any day,

- What is the probability that both Tom and John choose Cornflakes?
- What is the probability that neither Tom nor John choose Shredded Wheat?
- What is the probability that neither Tom nor John choose Cornflakes or Shredded Wheat?

#### Approach

Part (a)This is about mutually exclusive events. These are events that cannot occur at the same time. In this case only one colour ticket can be drawn first. If, for example, a salmon ticket is drawn that excludes the possibility of any other colour being drawn. You need to know that P(X) is the probability of X happening. Probabilities are often expressed as fractions so you may have to remember “fractions, decimals and percentages”.

Part (b) This is about Expected Frequency.

Expected Frequency = Probability of an Event Occurring x Number of Trials

In this example, Event Occurring = A customer buying a loaf of bread and the number of trials = The number of customers entering the supermarket.

Part (c) This is about Relative Frequency (could be referred to as Estimated or Experimental Probability) and theoretical probability.

Relative Frequency = no. successful trials/ total no. of trials.

Theoretical Probability = the probability that a certain outcome will occur, as determined through reasoning or calculation. So in this question the theoretical probability that the roulette wheel ball will land in any slot (assuming the wheel is not biased) is 1/37 as there are 37 slots (1 through 36 plus the “0” slot).

The more trials that are carried out the more you would expect the Relative Frequency to approximate to the Theoretical Probability. For example if the tossed a coin 3 times you might not be too surprised to get a Relative Frequency for “Heads” of 3/3 = 1.0 but if you tossed a coin a hundred coins you would find (assuming that the coin is not biased) that the Relative Frequency would come close to the Theoretical Probability of 0.5.

Part (d) This is about independent events. In this question Tom’s and John’s choices are independent of each other. To calculate the probability of two independent events occurring you multiply their probabilities. So if the chance of Y is 1/3 and the chance of Z is 2/3, then the probability of Y and Z occurring is:

P(Y) = 1/3 X P(Z) = 2/3 = 1/3 x 2/3 = 2/9

#### Answer

- a) The probability of a Pink ticket being drawn is

1 −(1÷3 + 5/12 + 1/4) Find the common denominator = 1– (4÷12 + 5/12 + 3/12) = zero!

The number of each colour tickets sold is:

Taupe = 1/3 x 4,800 = 1,600

Salmon = 5/12 x 4,800 = 2,000

Beige = 1/4 x 4,800 = 1,200

Pink none sold

Total 1,600 + 2,000 + 1,200 = 4,800

**Tip– with these sort of questions it’s always good to check that your calculations add up to the expected total.**

- b) The number of customers that may be expected to buy a loaf of bread is:

3,200 x 0.45 = 1,440

- c) The completed table is as follows:-

The theoretical probability of the ball landing in the “0” slot = 1/37 = .027. You would expect that the Relative Frequency would approximate to this and that it would become closer to the theoretical probability with more trials or spins. In fact the relative frequency after 1,000 spins is 0.045, which is considerably higher than the theoretical probability. So it seems possible that the wheel is biased to give a “0” result.

- d) The probability that both Tom and John choose cornflakes is

0.4 x 0.8 = 0.32 (32%)

The probability that neither Tom nor John choose Shredded Wheat is

(1– 0.50) x (1– 0.1) = 0.5 x 0.9 = 0.45 (45%)

The probability that neither Tom nor John will choose Cornflakes or Shredded Wheat is:

(1 — 0.4 — 0.5) x (1– 0.8 — 0.1) = 0.1 x 0.1 = 0.01 (1%)

Tree Diagram Question and Answer to follow.…