ATOT, help me with my Stats homework (again)

[Shadow Conception]

Please? It's probability.

There are two traffic lights on the route used by a certain individual to go from home to work. Let E denote the event that the individual must stop at the first light, and define the event F in a similar manner for the second light. Suppose that P(E) = 0.4, P(F) = 0.3, and P(E and F) = 0.15.

a) What is the probability that the individual needn't stop at either light?

-For this, I did P(~E and ~F) = (0.6)(0.7) = 0.42; the teacher's answer key says 0.45

b) What is the probability that the individual must stop at exactly one of the two lights?

-For this, I did P((E and ~F) or (~E and F)) = (0.4)(0.7) + (0.6)(0.3) = 0.46; the teacher's answer key says 0.4

c) What is the probability that the individual must stop just at the first light?

-For this, I did P(E and ~F) = (0.4)(0.7) = 0.28; the teacher's answer key says 0.25

I can't figure out what on earth I'm doing wrong. Thanks in advance for any help.

 

[us3rnotfound]

Huh, I would have come up with the same solutions as you, no joke.

 

[Kirby]

Wait, if P(E) = 0.4 and P(F) = 0.3, shouldn't P(E and F) be 0.12, not 0.15?

 

[octopus41092]

Wait, if P(E) = 0.4 and P(F) = 0.3, shouldn't P(E and F) be 0.12, not 0.15?

This. But other than this the reasoning seems correct. I would have done it the same way. My guess is you're misreading the question.

 

[KoolAidKid]

This. But other than this the reasoning seems correct. I would have done it the same way. My guess is you're misreading the question.

Only if E and F are independent, which they aren't.

 

[esun]

(a)

P(Doesn't need to stop at either light) = 1 - P(Needs to stop at either light)
P(Needs to stop at either light) = P(E or F) = P(E) + P(F) - P(E and F) = 0.4 + 0.3 - 0.15 = 0.55

=> P(Doesn't need to stop at either light) = 1 - 0.55 = 0.45

(b)

P(must stop at exactly one of the two lights) = P(must stop at either light) - P(must stop at both lights) = P(E or F) - P(E and F) = 0.55 - 0.15 = 0.4

(c)

P(must stop at just the first light) = P(must stop at either light) - P(must stop at the second light) = 0.55 - 0.3 = 0.25

EDIT: And your problem is that you are assuming the events are independent when they're not.

Think of it in terms of probability spaces (or in simpler terms, sets). There is an event E and an event F and overlap by some amount. The event E has size 0.4 while event F has size 0.3, and they overlap (E and F) by 0.15.

If you draw this out like a Venn diagram, you should be able to deduce the solutions easily. Shade the region the probability represents, figure out an expression for that region, and plug in the given values.

 

[Shadow Conception]

Wait, that's completely right, damn. I was treating it as an independent probability problem.

Seriously, thanks a shitload guys!

 

[nonameo]

shitload

I think this unit needs to be standardized.