If we are asked to find probabilities that are inequalities (and not equal to) then we can look up the values in our tables, but only if it is the form of P(X≤x). If it is not in that format we have to transform it using the following rules:
P(X<x) = P(X≤(x-1))

P(X>x) = 1 - P(X≤x)

P(X≥x) = 1 - P(X≤(x-1))

Once we have our probability in the correct format it is a matter of looking up the value in the tables.

**Example 1**

X∼B(20,0.15) find P(X≤5)

From the tables: 0.9327 (this means that the probability than the event X will occur less than or including 5 times is 93.27%)

**Example 2**

X∼B(40,0.05) find P(X<7)

P(X<7) = P(X≤6)

From the tables: 0.9966

**Example 3**

X∼B(30,0.5) find P(X>15)

P(X>15) = 1 - P(X≤15)

= 1 - 0.5722 (from the tables)

= 0.4278

**Example 4**

X∼B(10, 0.4) find P(X≥8)

P(X≥8) = 1 - P(X≤7)

= 1 - 0.9877 (from the tables)

= 0.0123

The expectation, E(X); average;mean, of a binomial distribution is np.

E(X) = np

**Example 1**

Find the expectation of the distribution where X∼B(20,0.2).

E(X) = 20*0.2

E(X) = 4 (this means that for a binomial distribution of 20 items or individuals, with a 20% probability of success, we expect 4 items/individuals to be successful)

**Example 2**

The expectation of a distribution equals 5, X∼B(n,0.5), find n.

E(X) = np

5 = 0.5n

n = 10

Therefore X∼B(10,0.5) to give an expectation of 5

The variance, σ^{2}, is given as npq.

Var(X) = np(1-p)

**Example 1**

Find the variance of the distribution where X∼B(10,0.3)

Var(X) = np(1-p)

Var(X) = 10*0.3*0.7

Var(X) =2.1

**Example 2**

The variance of a distribution equals 5, X∼B(25,p), find the possible values of p.

Var(X) = np(1-p)

5 = 25p(1-p)

1/5 = p - p^{2}

p^{2} - p + 1/5 = 0

Solve by simultaneous equation

p = 0.7236, or p = 0.2764

**NB:** Remember that E(aX + b) = a*E(X) + b; and that Var(aX + b) = a^{2}Var(X).

**Poisson Distribution**

We can approximate a binomial distribution with a Poisson distribution if the following conditions are met:

n is large (greater than 50 usually)

p is small (usually less than 0.3)

Because Poisson looks at the mean value, we have to calculate E(X) for the binomial and this is the mean value for our Poisson distribution.

If X∼B(n,p) and the suitable conditions are met then X≈Y∼Po(np)

**Example 1**

X∼B(100, 0.1), using a suitable approximation find P(X≥6)

X≈Y∼Po(10)

P(X≥6) = 1 - P(X≤5)

= 1 - 0.0671(Poisson tables)

= 0.9329

Therefore the probability that X is greater than or equal to 6 is approximately 93.29%

To find out more about the Poisson distribution click here.

We can approximate a binomial distribution with a normal distribution if the follow conditions are met:

n is large (greater than 50 usually)

p is close to 0.5

We can approximate the binomial with a normal to create Y∼N(np, np(1-p)^{2})

Hence X∼B(n,p)≈Y∼N(np, np(1-p)^{2}), we also have to do a continuity correction, because a normal distribution is continuous whereas a binomial distribution isn't. If we have to find P(X<80) then we don't want to include 80 in our search so we find P(X<79.5), for P(X>80) we find P(X>80.5), for P(X≤80) we find P(X<80.5) and for P(X≥80) we find P(X>79.5).

We would then solve normally using our normal distribution process.

**Example 1**

X~B(100, 0.5), use a suitable approximation to find P(X≤52)

X≈Y~N(50,25) and we want to find P(X≤52)

Our z score is calculated using:

z = (X-μ)/σ

Our continuity correction means we will take x as 52.5 because we want to include 52 in our calculation.

Hence z = (52.5 - 50)/5

Which gives a z value of 0.5. Looking this value up in our tables we find that P(X≤52)=0.6915

**Questions**
- If the following can be modelled as a Binomial distribution then state the distribution: a biased coin is tossed in the air, the probability of throwing a head is 0.4.
- If the following can be modelled as a Binomial distribution then state the distribution: a car passes a school 7 times in 1 day.

- X∼B(10,0.2) Find P(X=8)
- X∼B(15,0.1) Find P(X=20)
- X∼B(10,0.5) Find P(X=0)
- X∼B(8,0.4) Find P(X=2)
- X∼B(15,0.4) Find P(X=5)
- X∼B(20,0.15) Find P(X≤3)
- X∼B(25,0.01) Find P(X≤2)
- X∼B(50,0.5) Find P(X≤23)
- X∼B(10,0.2) Find P(X≤5)
- X∼B(10,0.2) Find P(X≥5)
- X∼B(10,0.2) Find P(X>5)
- X∼B(10,0.2) Find P(X<5)
- X∼B(25, 0.08) Find P(X≥7)
- X∼B(30,0.1) Find P(X<9)
- X∼B(40,0.2) Find P(X>10)

**Answers**
- X∼B(n,0.4) where X is the probability that a heads is thrown
- This cannot be modelled as a binomial, but can be done so as a Poisson
- 0.000073728
- 0 - 20 is outside the range
- 0.0010- This can be found by looking in the tables for P(X≤0)
- 0.2090
- 0.1859
- 0.6477
- 0.9980
- 0.3359
- 0.9936
- 0.0328 - 1-P(X≤4)
- 0.0064 - 1-P(X≤5)
- 0.9672 - P(X≤4)
- 0.0028
- 0.9980
- 0.1608