Practice: events and the rules of probability

Probability, grade 11 (5 units) Lesson 4 of 16Practice

Practice: events and the rules of probability

1

A blue die and a red die are thrown.
$\displaystyle A$ – “both dice show the same number”; $\displaystyle B$ – “the sum of the two dice is 8”.
Mark in the table the outcomes of the event $\displaystyle A\cup B$ .

a

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

b

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

/ – 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

2

A café recorded the orders of 200 customers.
120 customers ordered coffee, 70 ordered a pastry, and 40 ordered both.
A customer is chosen at random.

a

What is the probability that the customer ordered coffee or a pastry?

/ – 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

b

What is the probability that the customer ordered exactly one of the two?

/ – 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

c

What is the probability that the customer ordered neither coffee nor a pastry?

/ – 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

3

$\displaystyle {P(A)}={0.45}$ , $\displaystyle {P(B)}={0.35}$ .
The probability that neither $\displaystyle A$ nor $\displaystyle B$ happens is $\displaystyle 0.3$ .

a

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

b

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

/ – 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

c

What is the probability that exactly one of the two events happens?

/ – 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

4

$\displaystyle A$ and $\displaystyle B$ are disjoint events.
The probability of $\displaystyle B$ is $\displaystyle 0.2$ more than the probability of $\displaystyle A$ .
$\displaystyle {P(A\cup B)}={0.9}$ .

a

Write $\displaystyle {P(A)}={x}$ , and find $\displaystyle x$ .

b

Work out $\displaystyle P(\overline{A})$ .

/ – 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

5

A rectangular lawn measures 40 metres by 30 metres.
A sprinkler on the lawn waters a circle of radius 10 metres, all of it inside the lawn.
A raindrop falls on the lawn at random.

What is the probability that it falls inside the circle the sprinkler waters? Round to three decimal places.

/ – 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

6

A drawing pin is thrown 400 times.
It lands point up 252 times.

a

Estimate the probability that the pin lands point up.

/ – 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

b

The pin is thrown 1000 more times. About how many times will it land point up?

/ – 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

7

Yoav says: “If $\displaystyle {P(A)+P(B)}={1}$ , then $\displaystyle A$ and $\displaystyle B$ are complementary events.”
Show with an example of throwing a die that Yoav is wrong.

Formulas: probability, grade 11 (5 units)

The rules of probability

Equally likely outcomes – $\displaystyle k$ the outcomes in the event $\displaystyle A$ , $\displaystyle n$ the outcomes in the sample space
$\displaystyle P(A)=\frac{k}{n}$
Every event; $\displaystyle {P(\emptyset)}={0}$ , and $\displaystyle {P(U)}={1}$ for the sample space $\displaystyle U$
$\displaystyle 0\le P(A)\le1$
An event contained in another event
$\displaystyle P(A)\le P(B)$ $\displaystyle A\subseteq B$
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 $\displaystyle B$
$\displaystyle P(A\cup B)$ $\displaystyle =$ $\displaystyle P(A)+P(B)-P(A\cap B)$
Disjoint events – no two of them have an outcome in common
$\displaystyle P(A_1\cup A_2\cup\dots\cup A_n)=P(A_1)+P(A_2)+\dots+P(A_n)$

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
$\displaystyle P(A\cap B)$ $\displaystyle =$ $\displaystyle P(B)\cdot P(A/B)$
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 $\displaystyle A$
$\displaystyle P(A/B)$ $\displaystyle =$ $\displaystyle P(A)$
The law of total probability – the paths that end in $\displaystyle A$ , added up
$\displaystyle P(A)=P(B)\cdot P(A/B)+P(\overline{B})\cdot P(A/\overline{B})$

Counting and Bernoulli's formula

$\displaystyle n$ factorial; $\displaystyle {0!}={1}$
$\displaystyle n!=1\cdot2\cdot3\cdot\dots\cdot n$
Ordered selections – $\displaystyle k$ different objects out of $\displaystyle n$ , one after another
$\displaystyle n\cdot(n-1)\cdot\dots\cdot(n-k+1)=\frac{n!}{(n-k)!}$
The binomial coefficient – the ways to choose $\displaystyle k$ objects out of $\displaystyle n$ , order not counted
$\displaystyle \binom{n}{k}=\frac{n!}{k!(n-k)!}$
Bernoulli’s formula – the probability of $\displaystyle k$ successes in $\displaystyle n$ independent trials, with the probability of success $\displaystyle p$ in each
$\displaystyle P_n(k)=\binom{n}{k}p^k(1-p)^{n-k}$
At least one success in $\displaystyle n$ trials – by the complement
$\displaystyle 1-P_n(0)=1-(1-p)^n$