Practice: probability

Statistics and probability, grade 10 (4 units) Lesson 19 of 20Practice

Practice: probability

$\displaystyle {P(A)}={0.5}$ , $\displaystyle {P(B)}={0.4}$ , $\displaystyle {P(A\cap B)}={0.15}$ .

Work out $\displaystyle P(A\cup B)$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

What is the probability that neither of the two events happens?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

What is the probability that $\displaystyle A$ happens and $\displaystyle B$ does not?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

300 residents of a town were asked how they get to work or to their studies:

	By bus	By bike	By car	Total
Young	60	45	15	120
Adults	50	20	110	180
Total	110	65	125	300

A resident is chosen at random.

What is the probability that the resident comes by bike?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

What is the probability that the resident is young and comes by car?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

You met a young resident of the town. What is the probability that the resident comes by car?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

You met a resident who comes by car. What is the probability that the resident is young?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Why are the answers to parts c and d different, although both are about “young” and “by car”?

A factory has two machines.
Machine A makes 60% of the parts, and machine B 40%.
5% of machine A’s parts are faulty, and 10% of machine B’s.
A part is chosen at random.

Write the probabilities on the branches and each path’s probability.

What is the probability that the part is faulty?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The part is faulty. What is the probability that it was made by machine A?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Three boxes, $\displaystyle A$ , $\displaystyle B$ and $\displaystyle C$ , hold black and white balls:

box $\displaystyle A$ – two black balls and three white;
box $\displaystyle B$ – three black balls and two white;
box $\displaystyle C$ – four black balls and one white.

A box is chosen at random, and one ball is drawn from it at random.

What is the probability of drawing a white ball?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A white ball is known to have been drawn. What is the probability that it was drawn from box $\displaystyle B$ ?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The ball is put back.
Two balls are drawn at random from box $\displaystyle C$ , one after the other, without putting back.

What is the probability that after the drawing no white ball is left in box $\displaystyle C$ ?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Bag A holds seven yellow scarves and five red ones.
Bag B holds ten scarves: some yellow, and the rest red.
One scarf is drawn at random from bag A and one from bag B.
The probability that both scarves are yellow is $\displaystyle \frac{7}{40}$ .

How many yellow scarves are there in bag B? Call the number of yellow scarves in bag B $\displaystyle x$ , and write an equation.

Each scarf is put back in its bag, and again one scarf is drawn from bag A and one from bag B.
The scarves drawn are known to be of different colours.

What is the probability that the scarf from bag B is yellow?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Each scarf is put back in its bag again.
A bag is chosen at random, and two scarves are drawn from it at random, without putting back.

What is the probability that both scarves are red?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A fair die is thrown once.
$\displaystyle A$ – a number less than 3 comes up.
$\displaystyle B$ – an odd number comes up.

Work out $\displaystyle P(A\cap B)$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Are $\displaystyle A$ and $\displaystyle B$ independent? Explain.

Yossi plays three games of backgammon, one after the other. In each game he wins or loses (there are no draws).
If Yossi won a game, the probability that he wins the next one too is $\displaystyle p$ .
If he lost a game, the probability that he loses the next one too is $\displaystyle p$ .
$\displaystyle p>0.5$ .

Yossi is known to have won the first game. Express in terms of $\displaystyle p$ the probability that he loses the second game and wins the third.

Yossi is known to have won the first game, and the probability that he wins the third is $\displaystyle \frac{13}{25}$ . Write an equation and find $\displaystyle p$ .

The probability that Yossi wins all three games is $\displaystyle 0.144$ . Using the $\displaystyle p$ you found, find the probability that he wins the first game. Call it $\displaystyle q$ .

2% of the people in a population have a certain disease.
A test for the disease is positive in 95% of those who have it.
The test is also positive in 10% of those who are healthy.
A person is chosen at random and tested.

Write the probabilities on the branches and each path’s probability.

What is the probability that the test is positive?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The test came out positive. What is the probability that the person is ill? Round to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Most of those whose test is positive are healthy. Explain how this can be, when the test finds 95% of those who are ill.

Formulas: statistics and probability, grade 10 (4 units)

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Presenting data

Relative frequency – $\displaystyle f$ a value’s frequency, $\displaystyle n$ the number of all the data
$\displaystyle \frac{f}{n}$
Relative frequency in percent
$\displaystyle \frac{f}{n}\cdot100\%$
The angle of a slice in a pie chart
$\displaystyle \frac{f}{n}\cdot360^{\circ}$

Measures of centre and position

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
The mean from a frequency table – $\displaystyle x_1, \dots, x_k$ the values, $\displaystyle f_1, \dots, f_k$ their frequencies
$\displaystyle \bar{x}=\frac{x_1f_1+x_2f_2+\dots+x_kf_k}{f_1+f_2+\dots+f_k}$
A weighted mean – $\displaystyle w_1, \dots, w_k$ the weights of the values $\displaystyle x_1, \dots, x_k$
$\displaystyle \bar{x}=\frac{w_1x_1+w_2x_2+\dots+w_kx_k}{w_1+w_2+\dots+w_k}$
The mean of data in classes – $\displaystyle m_1, \dots, m_k$ the classes’ middles, $\displaystyle f_1, \dots, f_k$ their frequencies
$\displaystyle \bar{x}\approx\frac{m_1f_1+m_2f_2+\dots+m_kf_k}{f_1+f_2+\dots+f_k}$
The median’s place in an ordered list of $\displaystyle n$ data
$\displaystyle \frac{n+1}{2}$
The lower decile’s place in an ordered list; the upper decile’s – $\displaystyle 0.9\cdot n$
$\displaystyle 0.1\cdot n$

Measures of spread

The range – the greatest value minus the least
$\displaystyle x_{max}-x_{min}$
The variance – the mean of the squared deviations from the mean
$\displaystyle \sigma^2=\frac{(x_1-\bar{x})^2+(x_2-\bar{x})^2+\dots+(x_n-\bar{x})^2}{n}$
The standard deviation – the square root of the variance
$\displaystyle \sigma=\sqrt{\sigma^2}$
The variance from a frequency table
$\displaystyle \sigma^2=\frac{(x_1-\bar{x})^2f_1+\dots+(x_k-\bar{x})^2f_k}{f_1+\dots+f_k}$

The rules of probability

Equally likely outcomes – $\displaystyle k$ the number of outcomes in the event $\displaystyle A$ , $\displaystyle n$ the number of outcomes in the sample space
$\displaystyle P(A)=\frac{k}{n}$
The probability of any event; $\displaystyle {P(\emptyset)}={0}$ , and $\displaystyle {P(U)}={1}$ for the sample space $\displaystyle U$
$\displaystyle 0\le P(A)\le1$
The complement $\displaystyle \overline{A}$ – $\displaystyle A$ does not happen
$\displaystyle P(\overline{A})$ $\displaystyle =$ $\displaystyle 1-P(A)$
The union – $\displaystyle A$ or $\displaystyle B$ (or both); $\displaystyle A\cap B$ – the intersection, both $\displaystyle A$ and $\displaystyle B$
$\displaystyle P(A\cup B)$ $\displaystyle =$ $\displaystyle P(A)+P(B)-P(A\cap B)$
The union of disjoint events
$\displaystyle P(A\cup B)$ $\displaystyle =$ $\displaystyle P(A)+P(B)$ $\displaystyle A\cap B=\emptyset$

Conditional probability and independence

The probability of $\displaystyle A$ given $\displaystyle B$
$\displaystyle P(A|B)=\frac{P(A\cap B)}{P(B)}$
A path in a tree – the product of the probabilities on its branches
$\displaystyle P(A\cap B)$ $\displaystyle =$ $\displaystyle P(B)\cdot P(A|B)$
The law of total probability – a weighted mean of the conditional probabilities
$\displaystyle P(A)=P(B)\cdot P(A|B)+P(\overline{B})\cdot P(A|\overline{B})$
Bayes’ formula – from $\displaystyle P(B|A)$ to $\displaystyle P(A|B)$
$\displaystyle P(A|B)\cdot P(B)$ $\displaystyle =$ $\displaystyle P(B|A)\cdot P(A)$
Independent events
$\displaystyle P(A\cap B)$ $\displaystyle =$ $\displaystyle P(A)\cdot P(B)$
Independent events – knowing that $\displaystyle B$ happened does not change the probability of $\displaystyle A$
$\displaystyle P(A|B)$ $\displaystyle =$ $\displaystyle P(A)$