Approximations of the Binomial & Poisson Distributions |

**Approximating Binomial Probabilities**

Under certain sample size and proportion conditions, we can use the **Normal Distribution ( z-values) to approximate** sample statistics and their probabilities from a discrete proportional distribution like the Binomial.

When the sample size *n* is sufficiently large and

if 5 and 5,

the distribution of a proportion is **approximately Normal**.

We use these formulas for the mean and standard deviation:

The **Normal Distribution is continuous**, so any probability value in it represents the AREA under the curve on the given ** interval**. The Normal distribution, gives us the probability that

Should we need to find the probability that *x* is exactly equal to a given constant, say 7, we find the probability that *x* lies between 6.5 and 7.5.

The **Binomial distribution is discrete**, so to find the probability that the** random variable** takes on a specific value, we must assume that value extends over the **interval from 0.5 below it to 0.5 above** it. So, if we want the probability that 28, we find the probability that 27.5.

** Continuity Correction**:

- In order to apply the properties of a

we represent every

- To find

- To find

- To find

- To find , we find .

We represent *x* = 1 with the interval

**Example:**

- The Wayward Inn, a 300-room resort has experienced an 85% occupancy rate in the month of January for the past 10 years. The management is questionning whether this has changed so before they commit a lot of money to redecorating, they need to know:

a) the probability that **at least 260 rooms** will be occupied next January.

Since *n* = 300, and *p* = 0.85, we can see that 5 and 5,

we'll use the Normal Approximation of this Binomial situation.

Since we want , we'll set *x* = 259.5 and find .

**Solution:**

a)

- Now we need

The probability that **at least 260 rooms **will be occupied next January is **23.27%**

b) the probability that **fewer than 240 rooms** will be occupied next January?

**Solution:**

- Since we want , we'll set

- In this case, so we find

- Now we need P(

P(*z* < – 2.51) = 0.5000 – 0.4940 = 0.0060

The probability that **less than 240 rooms** will be occupied next January is **0.6%**.

**Note:** how we used **2****59.5** for **260** and **239.5** for **240** using the continuity correction factor.

**Approximating Poisson Probabilities**

We generally use the Poisson Distribution to **count** the number of **successes** over a **fixed interval** of time or within a specified region. For example: the number of murders recorded in Detroit in a month, or the number of cars parked at the airport during a 48 hour period.

The mean µ = *np*, as with the Binomial Distribution.

The standard deviation

The formula for calculating Poisson probabilities is:

where µ = the mean or expected number of successes,

*e* = the constant (2.718..), and *x* = the number of successes observed or counted.

**Note:** many statistics books use (lambda) instead of l for the expected mean value.

We can approximate the values of a Poisson Distribution

with the Normal Distribution when

*n* - the sample size is large, ...... *p* - the probability of a success is small,

and µ > 3, ...... with µ = *np*.

As with the Binomial Approximation, we use **continuity correction**.

**Example:** During the ski season, the medical clinic at Mont Tremblant Village treats an average of 3 patients with a broken bone per week, during the 8-hour day shift. If the number of patients per week with a broken bone approximates a Poisson distribution, find the probability that next week, the clinic will treat between 2 and 5 (inclusive) such patients. Use both the Poisson Distribution Table and the Normal Approximation, then compare the results.

**Solution: **

- We want =

- Using the table: = 0.2240 + 0.2240 + 0.1680 + 0.1008 =

- Using the Normal Approximation with

we set

and

- Since

- 0.3078 + 0.4251 =

**Practice**

Use the **Normal Approximation of the Binomial Distribution** with **Continuity Correction**

for questions **1 to 6**. Make a diagram, state µ and .

1) Records show that 25% of the cars manufactured at a certain plant need repairs in the

first 3 years of operation. If the IBM Corporation buys 90 of these cars,

find the probability that:

- a) in the first 3 years of operation,

- b)

2) The yearly number of major earthquakes in the world is a random variable having __approximately__ the normal distribution with µ = 20.8 and s = 4.5.

- Find the probabilities that in any given year, there will be:

- a) exactly 18 major earthquakes;

- b) at least 18 major earthquakes;

- c) at most 16 major earthquakes.

3) Research shows that a 3-foot high elm tree we transplant in the spring, has a 40% chance of surviving its first winter. If we transplant 50 such trees:

- a) What is the probability that 25 or more of them survive their first winter?

- b) What is the probability that between 18 and 23 of them survive their first winter?

4) Tests have shown that a new allergy drug is effective in 90% of the patients taking it. If the drug is administered to 80 allergy sufferers, what is the probability that it will be effective for at least 70 of them?

5) A certain contraceptive device is effective 90% of the time if used correctly.

If the device is used 300 times:

- a) how many times should we expect it to

- b) what is the probability that it will fail 35 or more times?

6) Canada Post claims that 80% of the letters mailed in Montreal, destined for Vancouver will be delivered within 3 working days. If you mail 200 such letters, find the probability that:

- a) more than 150 of them will arrive within 3 working days.

- b) fewer than 148 of them will arrive within 3 working days.

- c) between 150 and 160 of them will arrive within 3 working days.

- d) What's the minimum number of letters that arrived on time if the probability of this event is 10% or more?

Use the **Normal Approximation of the Poisson Distribution** with **Continuity Correction**

for questions **7 and 8**. Make a diagram, state µ and .

7) A parking lot attendant says he parks on average 12 cars per hour. What is the probability that he will park more than 15 cars between 3 and 4 p.m. today?

8) The Royal Bank guichet at the corner of Cavendish and Somerled is used 15 times per hour on an average day. Find the probability that **more than 12** people will use this guichet between 4 and 5 p.m. today if today is considered an average day.

**Solutions**

1) µ = *np* = 22.5, and

- a) for

_________________________

b) for **between** 20 and 23 cars **inclusive**, we use *x = *19.5 for 20 and *x* = 23.5 for 23 (see part a)

so, .

- so

- so

_____________________________________________

2) µ = 20.8 and s = 4.5 (these were given)

- a) for exactly 18, using Continuity Correction, we find

- .

- _________________________

- b) for

When we

- _________________________

- c) for at most 16, we find

- and

__________________________________________

3) µ = 50(.40) = 20, and

- a) for , we use

- ____________________________

- b)

__________________________________________

4) µ = 80(.90) = 72, and

- for , we use

- ____________________________

5) µ = 300(.10) = 30, and

- a)

- ____________________________

- b) for , we use

__________________________________________

6) µ = 200(.80) = 160, and

- a) for , we use

- ____________________________

- b) for , we use

- ____________________________

- c) for , we use

- the probability that = 0.4686 + 0.0359 =

_________________________________________

7) µ = 12 (given), and

- for , we use

_________________________________________

8) µ = 15 (given), and

- for , we use

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