Binomial probabilities | CAIE 9709 Probability & Statistics 1 Notes

Chapters

Probability & Statistics 1

Mathematics9709

Exactly r successes can occur in many orders

Exactly 2 successes in 4 trials

one order

S F S F

probability p²q²

SSFFSFSFSFFSFSSFFSFSFFSS

P(X = 2) = ⁴C₂p²q² = 6p²q²

Every order with two successes has probability p²q². There are ⁴C₂ = 6 such orders, so their probabilities are added by multiplying by 6.

One particular order with $r$ successes and $n-r$ failures has probability $p^r q^{n-r}$ . There are $\binom nr$ different positions for those successes, so multiply by that count.

p_r=\binom nr p^r q^{\,n-r}

Symbols

$p_r$ = P(X = r), the probability of exactly r successes (none)
$\binom nr$ = number of ways to place r successes among n trials (none)
$q$ = failure probability, 1-p (none)

Key idea

The exponents must add to the total number of trials:

r+(n-r)=n

. This is a quick check on every exact term.

Worked example

Exactly 3 successes from B(10, 0.4)

P(X=3)=p_3

p_3=\binom{10}{3}(0.4)^3(0.6)^7

Answer

P(X=3)=0.215

Translate words into an event before calculating

Swipe left or right to read every column →

Words	Event	Fast calculation
exactly r	$X=r$	PD(r)
at most r	$X\le r$	CD(r)
fewer than r	$X\le r-1$	CD(r - 1)
more than r	$X>r$	$1-\mathrm{CD}(r)$
at least r	$X\ge r$	$1-\mathrm{CD}(r-1)$
from a to b inclusive	$a\le X\le b$	$\mathrm{CD}(b)-\mathrm{CD}(a-1)$
not r	$X\ne r$	$1-\mathrm{PD}(r)$

On a calculator, PD gives one bar, $P(X=r)$ . CD adds all bars from 0 to the entered value, so $\mathrm{CD}(r)=P(X\le r)$ . Menu names differ between models, but the endpoint meaning is the same.

Common mistake

“Fewer than 8” ends at 7, not 8. Write the event before opening the calculator.

Use the calculator quickly and still show the method

For a binomial total, calculator CD is usually the fastest and safest method. Write the distribution and the translated event first.

Worked example

Fewer than 8 successes

A player scores independently with probability 0.7 on each of 10 attempts. Let $X$ count the goals.

X\sim B(10,0.7)

“Fewer than 8” means $P(X\le7)$ . Enter 10, 0.7 and upper endpoint 7 in binomial CD.

P(X<8)=\mathrm{CD}(7)

A complete term makes the binomial method visible:

p_7=\binom{10}{7}(0.7)^7(0.3)^3

Answer

P(X<8)=0.617

(9709/52/F/M/24 Q2(a))

Examiner note

Calculator CD is the quickest way to get the value. A bare decimal does not show the method: this mark scheme required at least one correct full binomial term for its method mark.

Your turn9709/52/F/M/25 Q2(b) adapted [3 marks]

Let $X\sim B(12,0.55)$ . Find $P(X<10)$ .

Show worked answer

“Less than 10” means $X\le9$ . The fastest entry is binomial CD with upper endpoint 9.

P(X<10)=P(X\le9)

P(X\le9)=\mathrm{CD}(9)

An equivalent method is to subtract the three upper-tail probabilities. The exact-probability formula above supplies each term:

Let $T$ be the upper-tail probability.

T=P(X\ge10)

T=p_{10}+p_{11}+p_{12}

For example, the first full tail term is

p_{10}=66(0.55)^{10}(0.45)^2

P(X<10)=1-T

Answer

P(X<10)=0.958

Exactly r successes can occur in many orders

Exactly 2 successes in 4 trials

one order

S F S F

probability p²q²

SSFFSFSFSFFSFSSFFSFSFFSS

P(X = 2) = ⁴C₂p²q² = 6p²q²

Every order with two successes has probability p²q². There are ⁴C₂ = 6 such orders, so their probabilities are added by multiplying by 6.

One particular order with $r$ successes and $n-r$ failures has probability $p^r q^{n-r}$ . There are $\binom nr$ different positions for those successes, so multiply by that count.

p_r=\binom nr p^r q^{\,n-r}

Symbols

$p_r$ = P(X = r), the probability of exactly r successes (none)
$\binom nr$ = number of ways to place r successes among n trials (none)
$q$ = failure probability, 1-p (none)

Key idea

The exponents must add to the total number of trials:

r+(n-r)=n

. This is a quick check on every exact term.

Worked example

Exactly 3 successes from B(10, 0.4)

P(X=3)=p_3

p_3=\binom{10}{3}(0.4)^3(0.6)^7

Answer

P(X=3)=0.215

Translate words into an event before calculating

Swipe left or right to read every column →

Words	Event	Fast calculation
exactly r	$X=r$	PD(r)
at most r	$X\le r$	CD(r)
fewer than r	$X\le r-1$	CD(r - 1)
more than r	$X>r$	$1-\mathrm{CD}(r)$
at least r	$X\ge r$	$1-\mathrm{CD}(r-1)$
from a to b inclusive	$a\le X\le b$	$\mathrm{CD}(b)-\mathrm{CD}(a-1)$
not r	$X\ne r$	$1-\mathrm{PD}(r)$

On a calculator, PD gives one bar, $P(X=r)$ . CD adds all bars from 0 to the entered value, so $\mathrm{CD}(r)=P(X\le r)$ . Menu names differ between models, but the endpoint meaning is the same.

Common mistake

“Fewer than 8” ends at 7, not 8. Write the event before opening the calculator.

Use the calculator quickly and still show the method

For a binomial total, calculator CD is usually the fastest and safest method. Write the distribution and the translated event first.

Worked example

Fewer than 8 successes

A player scores independently with probability 0.7 on each of 10 attempts. Let $X$ count the goals.

X\sim B(10,0.7)

“Fewer than 8” means $P(X\le7)$ . Enter 10, 0.7 and upper endpoint 7 in binomial CD.

P(X<8)=\mathrm{CD}(7)

A complete term makes the binomial method visible:

p_7=\binom{10}{7}(0.7)^7(0.3)^3

Answer

P(X<8)=0.617

(9709/52/F/M/24 Q2(a))

Examiner note

Calculator CD is the quickest way to get the value. A bare decimal does not show the method: this mark scheme required at least one correct full binomial term for its method mark.

Your turn9709/52/F/M/25 Q2(b) adapted [3 marks]

Let $X\sim B(12,0.55)$ . Find $P(X<10)$ .

Show worked answer

“Less than 10” means $X\le9$ . The fastest entry is binomial CD with upper endpoint 9.

P(X<10)=P(X\le9)

P(X\le9)=\mathrm{CD}(9)

An equivalent method is to subtract the three upper-tail probabilities. The exact-probability formula above supplies each term:

Let $T$ be the upper-tail probability.

T=P(X\ge10)

T=p_{10}+p_{11}+p_{12}

For example, the first full tail term is

p_{10}=66(0.55)^{10}(0.45)^2

P(X<10)=1-T

Answer

P(X<10)=0.958