Binomial probability model describes the number of successes in a specified number of trials. You...
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Transcript of Binomial probability model describes the number of successes in a specified number of trials. You...
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Binomial probability model describes the number of successes in a specified number of trials.
You need:* 2 parameters (success, failure)* Number of trials, n* probability of success, p
The Binomial Probability Model
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Example:A cereal manufacturer puts cards of 3 famous athletes in its cereal boxes. 20% of the boxes contain pictures of Derek Jeter, 30% contain a picture of David Beckham, and the rest contain a picture of Serena Williams.
Suppose you want to know what the probability is of getting 2 Derek Jeter cards if you buy 5 boxes. Could you use a binomial probability model?
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We can use a binomial probability model because:1 – there are two outcomes (success – getting a Derek Jeter card, failure – not getting a Derek Jeter card.2 – there is a set number of trials (n = 5)3 – we know the probability of success (p = .2)
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Using the model:
Let X = the random variableLet n = 5Let p = .2 (the probability of 1 success)Let k = 2 (the number of successes)
2 successes in 5 trials means 2 successes & 3 failures.
The possible order in which the outcomes can occur are disjoint (e.g., if 2 successes came in the first 2 trials, they couldn’t come in the last 2).
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Each different order in which we can have k successes in n trials is called a combination. It can be represented by nCk (n choose k) or .
k
n
To figure out the number of combinations in our trial:
10220
1231212345
)!(!!
knk
nk
n
This means there are 10 ways to get 2 Derek Jeter pictures in 5 boxes.
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Next step
2048.)8(.)2(.10)2( 32 XPNbr of successes
Nbr of combinations
Probability of success k
Probability of failure
(n- k)
The probability of success in getting 2 Derek Jeter cards in 5 boxes of cereal is .2048.
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Another example: The count of X children with type O blood among 5 children whose parents carry genes for both the O and the A blood types is B(5, 0.25). Find P(X=3).
Note: B means binomial setting, n = 5, p = 0.25
knk ppk
nXP
)1()3(
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23 )25.1(25.3
5)3(
XP
23 )75(.)25)(.20()3( XP0879.0)3( XP
Note: 0! = 1
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One step further:
Suppose the number of electrical switches that fail inspection is B(10, .1). What is the probability that no more than 1 switch fails.
7361.)1(
3487.3874.)1(
)3487)(.1)(1()3874)(.1)(.10()1(
)9(.)1(.0
10)9(.)1(.
1
10)1(
)0()1()1(
10091
XP
XP
XP
XP
XPXPXP
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µ = np
σ )1( pnp
These formulas are only good for binomial distributions. They can’t be used for other discrete random variables.
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Example:Using the previous cereal example, in 100 boxes of cereal, how many Derek Jeter cards do you expect to find?
Step 1: E(x)= np = (100)(.2) = 20Step 2: Step 3: Summary: In 100 cereal boxes, we expect to find 20 Derek Jeter cards, with a standard deviation of 4 cards.
)8)(.2)(.100()1( pnp 4
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Rule of thumb: If np ≥ 10 and n(1-p) ≥ 10, Normal approximation can be used.
Ex: (from your book page 527) A survey asked a nationwide random sample of 2500 adults if they agreed or disagreed that “I like buying clothes but shopping is often frustrating and time consuming.” Suppose that in fact 60% of all adult U.S. residents would say “Agree” if asked the same question. What is the probabiltiy that 1520 or more would agree?
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Since both criteria for Normal Approximation are satisfied, we can use Normal distribution calculations.
µ = (2500)(.6) = 1500
σ = = 24.49)4)(.6)(.2500(
2061.7939.01)82.()1520(
49.2415001520
49.241500
)1520(
)49.24,1500(
ZPXP
XPXP
N
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Using the calculator1) Probability distribution function – given a discrete random variable X, the probability distribution function assigns a probability to each value of X. See page 520, example 8.7 for binompdf(n,p,x)2) Cumulative distribution function – given a random variable X, the cumulative distribution function of X calculates the sum of probability of obtaining at most X successes in n trials. See page 522 example 8.10 for binomcdf(n,p,x)3) Look at the Technology Tool Box pages 530 - 532