New Zealand
Level 8 - NCEA Level 3

# Estimating probabilities

Lesson

When examining probabilities for a discrete random variable (DRV), we're not always given the probabilities nicely summarised with a function or a table of values.

Instead, we may need to analyse collected data for relative frequencies and what we call point estimates.

To explain this further let's take a look at an example.

#### Exploration

One hundred and fifty families, each with three children, were asked how many female children they had. The results are tabulated below.

Number of Female Children Frequency
$0$0 $15$15
$1$1 $55$55
$2$2 $62$62
$3$3 $18$18

We can use this data and add a relative frequency column to our table. These relative frequencies have been rounded to two decimal places.

Number of Female Children Frequency Relative Frequency
$0$0 $15$15 $0.1$0.1
$1$1 $55$55 $0.37$0.37
$2$2 $62$62 $0.41$0.41
$3$3 $11$11 $0.12$0.12

Adding this column is the same as working out the probability of each occurrence, based on our data. If we were to let $X$X be the number of female children in a three child-family, then we can represent our findings as follows.

$x$x $0$0 $1$1 $2$2 $3$3
$P(X=x)$P(X=x) $0.1$0.1 $0.37$0.37 $0.41$0.41 $0.12$0.12

Examining our table we notice that the probabilities add to $1$1. In some occasions the probabilities will not add to $1$1 due to rounding. If we wanted to, we could represent our relative frequencies as fractions instead, which will always avoid this problem.

Our data does represent a DRV. The outcomes are discrete and the probabilities are positive and sum to one.

If we compare it with the theoretical probabilities for the same scenario we can see the probabilities are quite close.

$x$x $0$0 $1$1 $2$2 $3$3
Probabilities from data $P(X=x)$P(X=x) $0.1$0.1 $0.37$0.37 $0.41$0.41 $0.12$0.12
Theoretical probability $P(X=x)$P(X=x) $0.125$0.125 $0.375$0.375 $0.375$0.375 $0.125$0.125

If we look at $P(X=1)=0.37$P(X=1)=0.37 for our collected data, we can say that the point estimate for a family of three children having only one female child is $0.37$0.37.

#### Practice questions

##### Question 1

Of $3000$3000 students graduating as teachers from a particular university, $8%$8% graduate as Mathematics teachers.

1. How many graduated as Mathematics teachers from this university?

2. In the country, there are $6900$6900 new graduate teachers. How many are likely to be Mathematics teachers?

##### Question 2

A family study is examining families with three children.

Of $350$350 families studied, $201$201 families have only one male child.

1. What proportion of families in the study have only one male child?

2. Can the number of males in a three-child family be considered a discrete random variable?

Yes

A

No

B

Yes

A

No

B
3. Complete the table of values for the discrete random variable, where $X$X represents the number of male children in a three-child family in the overall population. Assume boys and girls are equally likely.

 $x$x $P\left(X=x\right)$P(X=x) $0$0 $1$1 $2$2 $3$3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
4. What proportion of families would you expect to have one male child?

##### Question 3

As part of a psychological experiment, a group of $8$8 people are given a multiple choice on a topic they know nothing about and are asked to answer all questions.

The number of questions they each answer correctly out of $10$10 are:

$2,3,3,3,1,3,4,1$2,3,3,3,1,3,4,1

1. On average, how many questions do they answer correctly?

2. On average, what proportion of questions do they answer correctly?

3. Based on these results, how many choices do they seem to have for each question?

### Outcomes

#### S8-4

Investigate situations that involve elements of chance: A calculating probabilities of independent, combined, and conditional events B calculating and interpreting expected values and standard deviations of discrete random variables C applying distributions such as the Poisson, binomial, and normal

#### 91586

Apply probability distributions in solving problems