Help with a simple probability problem

  • Thread starter GeoMike
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In summary, the probability of selecting exactly 8 women in a group of 20 motorcycle drivers is (20!/(12!*8!)) * (1/20)^8(19/20)^12 and the probability of selecting at least 8 women is the sum of the probabilities of exactly 8 through exactly 20 women.
  • #1
GeoMike
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Ok, this seems like an easy enough problem, but I'm stumped...

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If a motorcycle driver is selected at random from the US population, the probability of that driver being a male is 0.95
If 20 motorcycle drivers are selected at random from the population, what is the probability that exactly 8 will be women? What is the probability that at least 8 will be women?
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I've got that the chance of selecting a female is 1/20. And because the population is so large I can treat the events as independent.

I can easily find the probability of selecting 8 women in a row. What's throwing me off is the fact that there are 20 selections, so I know that the probability of selecting 8 total isn't going to be the same as selecting 8 in a row. I can't figure out how to account for the fact that 20 selections increases the probability of 8 women being selected.
I'm also thrown off by the at least 8 vs. exactly 8 part of the problem.

Any hints would be great. Thanks! :biggrin:
-GeoMike-
 
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  • #2
if event A has a probability p of occurring, and an event B has a probability 1-p of occurring, and one event or the other occurs n times, you can characterize the probability by

pk(1-p)n-k

where k is the number of times A happens (this assumes order counts. In your problem it doesn't, so you'll need to fix it for that)
 
  • #3
Ok, that makes sense. So, the probability of getting 8 females and 12 males (in a specific combination) would be:

(1/20)8(19/20)12

To account for the fact that any arrangement of 8 females and 12 males is acceptable, I'd just multiply that probability by the total number of combinations:

(20!/(12!*8!)) * (1/20)8(19/20)12 = P(exactly 8 women)

Right?

And to find P(at least 8 women) I'd just sum the probabilities of exactly 8 through exactly 20 women:

[tex]\sum_{i=8}^{20}{\frac{20!}{(20-i)!i!}(1/20)^{x}(19/20)^{20-x}}[/tex]

Right?

Thanks,
-GeoMike-
 
  • #4
of course, you'll have fewer things to sum of you do 0 to 7 females... and something about 1-p

Although, in this case, it's not that big a deal. But, if it was "at least 2 females", it'd be far quicker to find the probabilities of 0 females and 1 female, then subtracting from 1.
 

Related to Help with a simple probability problem

1. What is probability?

Probability is a measure of the likelihood that a certain event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

2. How do you calculate probability?

To calculate probability, divide the number of desired outcomes by the total number of possible outcomes. This can be expressed as a fraction, decimal, or percentage.

3. What is the difference between theoretical and experimental probability?

Theoretical probability is based on mathematical calculations and assumes that all outcomes are equally likely. Experimental probability is based on actual results from experiments or observations.

4. How do you represent probability?

Probability can be represented using fractions, decimals, percentages, or as odds (e.g. 1 in 5 chance). It can also be represented using a probability scale from 0 to 1, where 0 represents impossibility and 1 represents certainty.

5. How can probability be applied in real life?

Probability is used in many real-life situations, such as predicting weather patterns, determining the likelihood of winning a game, and calculating risk in insurance and finance industries. It is also used in scientific research to analyze and interpret data.

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