Question

Scooby-Doo has become very successful at fighting crime. He and Shaggy can successfully solve a case 80% of the time.

a) What is the probability that Scooby and Shaggy will not solve the first four cases before being successful? [2]

b) What is the probability that Scooby’s first three cases will be unsuccessful? [2]

Answer 1

@blockcolder @satellite73

Answer 2

since 20% is the fail rate and in fraction form it is 1/5 i'm thinking \[\frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5}\] feel free to correct me though @blockcolder @satellite73

Answer 3

I don't understand Chloro. what happened?

Answer 4

It's geometric type!

Answer 5

@jim_thompson5910

Answer 6

have you ever seen or worked with the geometric probability distribution before?

Answer 7

Not really. I've worked more on hypergeometric distributions than geometric distributions.

Answer 8

Well to answer this question, you would need to use the geometric distribution as the hypergeometric distribution is a lot like the binomial distribution (but no replacements are made)

Answer 9

Is the formula that I'm supposed to use in this document?

Answer 10

@jim_thompson5910

Answer 11

alright one sec while I read the attachment

Answer 12

It looks like you are using the formula you highlighted in bold on the left because that matches with the formula I'm looking at (the only difference is that q is 1 - p, but they are the same)

Answer 13

Part a)


Use the geometric probability distribution with p = 0.8 and k = 5


P(X = k) = (1-p)^(k-1)*p


P(X = 5) = (1-0.8)^(5-1)*0.8


P(X = 5) = (0.2)^(4)*0.8


P(X = 5) = 0.0016*0.8


P(X = 5) = 0.00128


So the probability of the first success on the fifth trial (ie they solve their first case on the fifth try) is 0.00128

Answer 14

P(x)= q^x p
= 0.20^4 x 0.80
= 0.00128

Answer 15

You nailed it. Nice job

Answer 16

Hey, thank you! :)

Answer 17

You're welcome. Want to take a stab at part b?

Answer 18

Sure!

Answer 19

P(x)= q^x p
=0.20^3 x 0.80
=0.0064

Answer 20

After looking at this part again in a bit more depth, I'm thinking that it's better modeled by the negative binomial probability distribution. Have you dealt with this distribution before?

Answer 21

hmm nevermind that last thought. The negative binomial distribution is used to count the number of events and not the order of the events, so that's not too useful.


lol sry, it's getting late over here and I'm falling asleep...

Answer 22

It's alright! Thanks for the help though!

Answer 23

How many cases would you expect Scooby and Shaggy to not solve before they are successful? [2] @satellite73

Answer 24

@jim_thompson5910

Answer 25

@jim_thompson5910

Answer 26

In this part, you're finding the expected value of the geometric distribution, which is (1-p)/p where p is the probaility of success.

Answer 27

probability*

Answer 28

1-0.80/ 0.80
= 0.20/0.80
= 0.25

@jim_thompson5910

Answer 29

you got it, very nice work

Answer 30

Thanks. Hey, do you know how the equation begins?

Answer 31

what do you mean?

Answer 32

Like for the formula, does it begin with "P(x)=" ?

Answer 33

oh, it's just E(X) or E[X]

Answer 34

to denote expected value

Answer 35

Oh ok. thanks!

Answer 36

yw

Answer 37

Oh, but does that answer the question?

How many cases would you expect Scooby and Shaggy to not solve before they are successful? [2]

I would expect Scooby and Shaggy to not solve 0.25 cases before being successful?

Answer 38

yes, that's exactly what the expected value finds

Answer 39

Oh ok.