Probability Distribution Practice

Q: What is the fastest way to find P(X\ge 1)?

Use the complement: P(X≥1)=1−P(X=0)P(X\ge 1)=1-P(X=0)P(X≥1)=1−P(X=0). Instead of adding every case from 111 upwards, you compute the single term P(X=0)=q nP(X=0)=q^{\,n}P(X=0)=qn and subtract it from 111. This is the standard “at least one”\text{``at least one''}“at least one” shortcut.

Q: How do I read the table when the Z-value is negative?

The standard normal curve is symmetric about 000, so P(Z<−a)=P(Z>a)P(Z<-a)=P(Z>a)P(Z<−a)=P(Z>a). Look up the positive value aaa in the table and use it directly. Always sketch the curve and shade the region you want before reading off a number.

Q: What are the mean and variance of a binomial distribution?

For X∼B(n,p)X\sim B(n,p)X∼B(n,p) the mean is μ=np\mu=npμ=np and the variance is σ2=npq\sigma^{2}=npqσ2=npq, where q=1−pq=1-pq=1−p. The standard deviation is σ=npq\sigma=\sqrt{npq}σ=npq; take the square root only at the very end.

How to use these practice questions

The six questions below rise in difficulty across the whole chapter, from a single binomial probability to finding an unknown mean of a normal distribution. Give yourself roughly five to eight minutes per question and work on paper first, writing every line the way you would in the real exam, the distribution and its parameters, the formula you are using, a clear substitution, then the final probability.

Keep the standard normal distribution table beside you for the normal questions, and resist the urge to peek.

Only once you have committed to a full answer should you open the solution and mark yourself line by line. When your working differs from ours, stop at the exact line where the two part company; that single step is almost always where the mark was lost.

Because Add Math is marked analytically, a correctly quoted formula and a correct $Z$ -value still earn credit even when the final rounding slips, so always write the formula before you substitute, and remember every probability must lie between $0$ and $1$ .

Six practice questions

Q1[3 marks]

A discrete random variable $X$ follows a binomial distribution with $n=5$ and $p=0.2$ , written $X\sim B(5,\,0.2)$ . Find $P(X=2)$ .

Show worked solution

Use the binomial formula $P(X=r)={}^{n}C_{r}\,p^{r}\,q^{\,n-r}$ , where $q=1-p=0.8$ . Substitute $n=5$ , $r=2$ :

P(X=2)={}^{5}C_{2}\,(0.2)^{2}\,(0.8)^{3}

=10\times 0.04\times 0.512=0.2048

Answer

$P(X=2)=0.2048$ . Check the pieces: ${}^{5}C_{2}=10$ , $(0.2)^{2}=0.04$ , $(0.8)^{3}=0.512$ ; their product is $0.2048$ , which lies between $0$ and $1$ as a probability must.

Q2[3 marks]

A discrete random variable $X$ follows $X\sim B(25,\,0.2)$ . Find the mean and the standard deviation of $X$ .

Show worked solution

For a binomial distribution the mean is $\mu=np$ and the variance is $\sigma^{2}=npq$ , with $q=1-p=0.8$ . First the mean:

\mu=np=25\times 0.2=5

Now the variance, then take the square root for the standard deviation:

\sigma^{2}=npq=25\times 0.2\times 0.8=4

\sigma=\sqrt{4}=2

Answer

The mean is $5$ and the standard deviation is $2$ . As a check, the standard deviation is smaller than the mean here, which is typical for a binomial with these parameters.

Q3[4 marks]

A discrete random variable $X$ follows $X\sim B(5,\,0.4)$ . Find $P(X\ge 1)$ .

Show worked solution

Adding $P(X=1)$ up to $P(X=5)$ is long, so use the complement: $P(X\ge 1)=1-P(X=0)$ . With $q=1-0.4=0.6$ :

P(X=0)={}^{5}C_{0}\,(0.4)^{0}\,(0.6)^{5}=(0.6)^{5}=0.07776

P(X\ge 1)=1-0.07776=0.92224

Answer

$P(X\ge 1)=0.9222$ (to four decimal places). The complement is the fast route whenever a question asks for $\text{``at least one''}$ : compute the single term $P(X=0)$ and subtract from $1$ .

Q4[4 marks]

A continuous random variable $X$ is normally distributed with mean $50$ and standard deviation $10$ , that is $X\sim N(50,\,10^{2})$ . Find $P(X>65)$ .

Show worked solution

Standardise with $Z=\dfrac{X-\mu}{\sigma}$ , converting $X=65$ to a $Z$ -value using $\mu=50$ and $\sigma=10$ :

Z=\frac{65-50}{10}=\frac{15}{10}=1.5

P(X>65)=P(Z>1.5)

From the standard normal distribution table, $P(Z>1.5)=0.0668$ .

Answer

$P(X>65)=0.0668$ . Because $65$ lies above the mean, the tail probability is well below $0.5$ , which is the sensible order of size.

Q5[4 marks]

A continuous random variable $X$ is normally distributed with mean $60$ and standard deviation $8$ , that is $X\sim N(60,\,8^{2})$ . Find $P(52<X<76)$ .

Show worked solution

Standardise both endpoints with $Z=\dfrac{X-\mu}{\sigma}$ , using $\mu=60$ and $\sigma=8$ :

Z_{1}=\frac{52-60}{8}=-1 \qquad Z_{2}=\frac{76-60}{8}=2

P(52<X<76)=P(-1<Z<2)

From the table $P(Z>2)=0.0228$ and $P(Z>1)=0.1587$ . Write the interval as $P(Z<2)-P(Z<-1)$ , and use the symmetry $P(Z<-1)=P(Z>1)$ :

P(-1<Z<2)=\big(1-0.0228\big)-0.1587

=0.9772-0.1587=0.8185

Answer

$P(52<X<76)=0.8185$ . The interval straddles the mean and covers a wide range, so a large probability is exactly what we expect.

Q6[5 marks]

A continuous random variable $X$ is normally distributed with mean $\mu$ and standard deviation $5$ . Given that $P(X>30)=0.1587$ , (a) find $\mu$ , and (b) hence find $P(X<20)$ .

Show worked solution

(a) First find the $Z$ -value for which $P(Z>z)=0.1587$ . From the standard normal table this is $z=1$ .

Now standardise $X=30$ and set its $Z$ -value equal to $1$ :

\frac{30-\mu}{5}=1 \;\Rightarrow\; 30-\mu=5 \;\Rightarrow\; \mu=25

(b) With $\mu=25$ and $\sigma=5$ , standardise $X=20$ , then use symmetry:

Z=\frac{20-25}{5}=-1

P(X<20)=P(Z<-1)=P(Z>1)=0.1587

Answer

$\mu=25$ and $P(X<20)=0.1587$ . This equals the given $P(X>30)$ , which makes sense: $20$ and $30$ are each $5$ units, one standard deviation, from the mean $25$ , one on each side, so their tail probabilities match.

How to mark yourself like an examiner

Add Math is marked analytically, which means marks are attached to steps, not only to the final number. When you check your own script, award yourself credit the way a marker would: look for the correct distribution, the right formula or $Z$ -value, and a clean final probability.

Method mark: did you quote the correct formula, the binomial $P(X=r)={}^{n}C_{r}\,p^{r}q^{\,n-r}$ , the mean $np$ and variance $npq$ , or the standardisation $Z=\dfrac{X-\mu}{\sigma}$ ?
Parameter mark: for a binomial, are $n$ , $p$ and $q=1-p$ correct, with the right power on each factor? For a normal, is $\sigma$ (not $\sigma^{2}$ ) used in the denominator?
For $\text{``at least one''}$ : did you use the complement $1-P(X=0)$ rather than a long sum?
For a normal probability: is the $Z$ -value correct, is the table read as a tail $P(Z>z)$ , and is symmetry $P(Z<-a)=P(Z>a)$ applied for negative $Z$ ?
Answer mark: is the final probability between $0$ and $1$ ? Any value outside that range is an immediate signal that a step went wrong.

How a teacher helps

Marking yourself is powerful, but it is hard to see your own blind spots. In a one-to-one lesson our teacher watches the exact line where a mark slips away, a variance used where the standard deviation was needed, a table tail read from the wrong side, or an $\text{``at least''}$ tackled the long way instead of by the complement, and corrects the habit on the spot.

Because our teachers are experienced, you work with someone who explains when to reach for the binomial and when for the normal. Lessons are taught in English, while SPM papers are set in both Malay and English, so we make sure the notation reads the same to you either way.

Get 1-to-1 help.

Book a Trial Class

Frequently asked questions

How do I decide between the binomial and the normal distribution?

Use the binomial when you have a fixed number of independent trials and you are counting how many succeed, a discrete whole-number count with $X\sim B(n,p)$ . Use the normal when the variable is a continuous measurement described by a mean and a standard deviation, $X\sim N(\mu,\sigma^{2})$ .

What is the fastest way to find

P(X\ge 1)

Use the complement: $P(X\ge 1)=1-P(X=0)$ . Instead of adding every case from $1$ upwards, you compute the single term $P(X=0)=q^{\,n}$ and subtract it from $1$ .

This is the standard $\text{``at least one''}$ shortcut.

How do I read the table when the

Z

-value is negative?

The standard normal curve is symmetric about $0$ , so $P(Z<-a)=P(Z>a)$ . Look up the positive value $a$ in the table and use it directly.

Always sketch the curve and shade the region you want before reading off a number.

What are the mean and variance of a binomial distribution?

For $X\sim B(n,p)$ the mean is $\mu=np$ and the variance is $\sigma^{2}=npq$ , where $q=1-p$ . The standard deviation is $\sigma=\sqrt{npq}$ ; take the square root only at the very end.

Source:SRC-DSKP-EN

Written by the spmaddmath.com.my editorial team.· Last updated 5 September 2026

Probability Distribution, Practice Questions

How to use these practice questions

Six practice questions

How to mark yourself like an examiner

How a teacher helps

Frequently asked questions

Ready to get started?