Cumulative probabilities 

​​In a nutshell

Cumulative probabilties for the binomial distribution are probabilities that satisfy a specific inequality.

Definition

The cumulative probability is the probability $$P(X \le x)$$​. It counts all of the probabilities up to and including the specific output $$x$$​.

Recall: the binomial distribution

The binomial distribution is represented as:

Finding the cumulative probability for the binomial distribution 

While it is possible to find cumulative binomial probabilities using a table, it is much easier to use a calculator.

calculator tip

$$\boxed{\begin{aligned}Binomial\space cd\\Data&: variable\\x&:\\N&:\\p&:\end{aligned}}$$​​

Note: Make sure to select the correct distribution! If you select binomial pd, you will get $$P(X=x)$$ and not $$P(X \le x)$$.

Example 1

If $$X \sim B(10,0.4)$$​, find the values of:

i) $$P(X \leq 5)$$​

ii) $$P(X< 5)$$

iii) $$P(X>5)$$​

iv) $$P(X \ge 5)$$​

i) Simply use the binomial cd function on your calculator with $$N=10,p=0.4,x=5$$:

$$\underline{P(X \leq 5 )=0.8338}$$​​

ii) To use the binomial cd function, the inequality has to be in the form $$P(X\le x)$$. Note that since the outputs are integers, $$X<5$$ is the same as $$X\leq 4$$. Hence:

​$$\underline{P(X<5)=P(X\leq4)=0.6331}$$​​

iii) Use the fact that probabilities add up to $$1$$:

​$$P(X\leq5)+P(X>5)=1\\\begin{aligned}P(X>5)&=1-P(X\leq5)\\&=1-0.8338\\&=0.1662\end{aligned}$$​​

​$$\underline{P(X >5)=0.1662}$$​​

iv) Again, use the fact that probabilities add up to $$1$$, and that $$X<5$$ is the same as $$X\leq4$$:

$$P(X<5)+P(X\geq5)=1\\\begin{aligned}P(X\geq 5)&=1-P(X<5)\\&=1-P(X\le 4)\\&=1-0.6331\\&=0.3669\end{aligned}$$​​

$$\underline{P(X\ge5)=0.3669}$$​​

Questions in context

You may be asked to find cumulative probabilities for contextual events. To do this, you need to be able to recognise certain phrases and relate them with the corresponding cumulative probability.

Example 2

A biased coin has a $$70\%$$ chance of turning up heads. If the coin is flipped $$8$$ times, what is the probability that:

i) The coin turns up heads no more than $$3$$ times.

ii) The coin turns up heads at least $$5$$ times.

First, define the random variable and identify the correct distribution.

Let $$X$$ be the number of times the coin turns up heads.

​$$X\sim B(8,0.7)$$​​

i) Associate the phrase "no more than $$3$$​" with an inequality.

"No more than $$3$$" is the same as $$X\leq3$$.

​$$\underline{P(X\leq 3)=0.0580}$$​​

ii) Associate the phrase "at least $$5$$" with an inequality. 

"At least $$5$$" is the same as $$X \ge5$$.

​$$\begin{aligned}P(X\geq5)&=1-P(X<5)\\&=1-P(X\leq4)\\&=1-0.1941\\&=0.8059\end{aligned}$$​​

​$$\underline{P(X\geq 5)=0.8059}$$​​