achiever math probability question

[nartnad]

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"If 1/3 of the cars driven by residents in a specific town are Japanese made, what is the probability of seeing at least one Japanese car out of every three automobiles on the road?"

I dont understand the logic here...why is the probability not (simply) one third?

the answer is:

"The probability of NOT seeing any Japanese car out of every three automobiles is (2/3)3 = 8/27. {Multiplication Rule}

Using complementary rule, the probability of seeing at least one Japanese car out of every three automobiles is therefore,

1 – (8/27) = 19/27"

 

[Streetwolf]

"If 1/3 of the cars driven by residents in a specific town are Japanese made, what is the probability of seeing at least one Japanese car out of every three automobiles on the road?"

I dont understand the logic here...why is the probability not (simply) one third?

the answer is:

"The probability of NOT seeing any Japanese car out of every three automobiles is (2/3)3 = 8/27. {Multiplication Rule}

Using complementary rule, the probability of seeing at least one Japanese car out of every three automobiles is therefore,

1 – (8/27) = 19/27"

First of all, because it says 'AT LEAST' one.
Second of all, because you are considering more than one car. Not only does that one car need to be Japanese, but the other two have to NOT be Japanese.

Since you want to know the probability of seeing at least 1 out of every 3, then you need to consider the cases:

You see 1 car out of 3.
You see 2 cars out of 3.
You see 3 cars out of 3.

Now you can either find the probabilities of all 3 of those happening and add them together... or you can find the probability of the only remaining scenario:

You see 0 cars out of 3.

How does that happen?

P(first car isn't Japanese) = 2/3
P(second car isn't Japanese) = 2/3
P(third car isn't Japanese) = 2/3

(2/3) * (2/3) * (2/3) = 8/27

So there's an 8/27 chance that you DON'T see a Japanese car. Since all 4 cases MUST total 1 (WHY?), you know that the probability you see AT LEAST one Japanese car is:

1 - (8/27) = 19/27.

===

FYI the odds of seeing EXACTLY one Japanese car:

(3 C 1) * (1/3)(2/3)(2/3) = 4/9. So even that isn't just 1/3. HOWEVER, of all the 4 cases described above, this case will have the highest probability of occurring since it matches the probability. As another example, if there was a 2/7 chance of seeing a Japanese car and you observed 7 cars, the scenario where you observe 2 Japanese cars out of 7 would have the highest probability of occurring out of all the possible scenarios with 7 cars.

 

[nartnad]

thanks for the tip. i love math, but hate probability.