IB DP Maths: AA HL復(fù)習(xí)筆記4.5.1 The Binomial Distribution
Properties of Binomial Distribution
What is a binomial distribution??
- A binomial distribution is a?discrete probability distribution
What are the important properties of a binomial distribution?
- The?expected number (mean)?of successful trials is
-
You are given this in the?formula booklet- Square root to get the?standard deviation
- The distribution can be represented visually using a vertical line graph
- If?p?is?close to 0?then the graph has a?tail to the right
- If?p?is?close to 1?then the graph has a?tail to the left
- If?p?is?close to 0.5?then the graph is?roughly symmetrical
- If?p?=?0.5?then the graph is?symmetrical
Modelling with Binomial Distribution
How do I set up a binomial model?
- Identify?what a?trial?is in the scenario
- For example: rolling a dice, flipping a coin, checking hair colour
- Identify?what the?successful outcome?is in the scenario
- For example: rolling a 6, landing on tails, having black hair
- Identify?the?parameters
- n?is the number of trials and?p?is the probability of success in each trial
- Make sure you?clearly state?what your?random variable?is
- For example, let be the number of students in a class of 30 with black hair
What can be modelled using a binomial distribution?
- Anything that satisfies the?four conditions
What can not be modelled using a binomial distribution?
- Anything where the number of trials is?not fixed?or is?infinite
- The number of emails received in an hour
- The number of times a coin is flipped until it lands on heads
- Anything where the outcome of?one trial affects?the outcome of the?other trials
- The number of caramels that a person eats when they eat 5 sweets from a bag containing 6 caramels and 4 marshmallows
- If you eat a caramel for your first sweet then there are less caramels left in the bag when you choose your second sweet
- Anything where there are?more than two outcomes?of a trial
- A person's shoe size
- The number a dice lands on when rolled
- Anything where the?probability of success changes
- The number of times that a person can swim a length of a swimming pool in under a minute when swimming 50 lengths
- The probability of swimming a lap in under a minute will decrease as the person gets tired
- The probability is?not constant
- The number of times that a person can swim a length of a swimming pool in under a minute when swimming 50 lengths
- The number of caramels that a person eats when they eat 5 sweets from a bag containing 6 caramels and 4 marshmallows
Exam Tip
- An exam question might involve different types of distributions so make it clear which distribution is being used for each variable
Worked Example
It is known that 8% of a large population are immune to a particular virus. Mark takes a sample of 50 people from this population. Mark uses a binomial model for the number of people in his sample that are immune to the virus.
a)? ? ? State the distribution that Mark uses.
b)? ? ? State two assumptions that Mark must make in order to use a binomial model.
c)? ? ? Calculated the expected number of people in the sample that are immune to the virus.
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