dddanny2006 wrote:Hey People
Please help me with this probability problem.I have attached the image here.
My doubt is --
When we solve to find the probability that Leo will hear a song he likes--Why cant 0.30*0.30*0.30 or 0.30+.30+.30 be the answer?
Please differentiate between the two.
Also,the following is the right way to solve this problem--
Method 1
Prob(Leo likes first station) +Prob(Leo doesnt like first station,but like second one)+Prob(Leo doesnt like first,second,but like the third)
=(0.30) + (0.30*0.70) + (0.30*0.30*0.70) = 0.657
There's another way to do this problem its
Method 2
P(No station playing the song)=1-(0.7*0.7*0.7)=0.657
What I dont understand is we directly multiply .7*.7*.7 in the second method,but when we solve using the first method we have the extra terms such as 0.70 etc..
Why is that??
Thanks
Dear
dddanny2006,
I'm happy to help with this.
First, here's a post about some probability rules:
https://magoosh.com/gmat/2012/gmat-math- ... ity-rules/
First of all, this expression ...
(0.30)*(0.30)*(0.30)
... doesn't work at all because Leo is not going to listen to three songs likes on three different stations. If he hears a song on Station A, he is going to stay there, and never get to stations B or C. In this scenario, Leo never listens to more than one song that he likes. We can only multiply probabilities if the events are
independent, and they are not: Leo hearing any song on station B explicitly depends on his not liking the song on station A.
Similarly, this expression ...
0.30 + 0.30 + 0.30
...doesn't work. This is trickier. It's true that he is looking for A or B or C, and it's true that the three are mutually exclusive. Adding only works for "or" probabilities in what we might call a side-by-side choice ---- all three possibilities are in front of me at once, and I am waiting to see which of the three arises. Think of rolling a die, for example. ---- When we have different events in a sequence, with earlier outcomes determining how the sequence plays out, then we can no longer use the simple "or" rule.
Then you ask about method 1 vs. 2.
Method #1 use the Generalized "AND" Rule. The Generalized "AND" Rule makes use of something called
conditional probabilities. We use the notation P(A|B), which is read "
the probability of A, given B." The conditional probability P(A|B) means the following: first, we are going to pretend that condition B is true; then, in the world in which B is true and can be taken for granted, what is the probability of A?
The Generalized "AND" Rule says that, for events that are not independent,
P(A and B) = P(A)*P(B|A)
That's what is used here. To figure out, say, the probability that Leo hears the song he like on B --- well, in this scenario, that depends on not hearing a song he likes on A. In order to get to the outcome of "likes the song on B", we have to pass through the condition of "doesn't like the song on A." Both must happen, which is why we multiply them.
In words, what happens is the following. For brevity, I will use the notation "Leo + A" to mean "Leo finds a song he likes on Station A." There are three distinct "paths" to success here.
Path one: (Leo + A)
or
Path two: (Leo does not + A) and (Leo + B)
or
Path three: (Leo does not + A) and (Leo does not + B) and (Leo + C)
Notice the events within paths are joined by "and" statements, so we multiply, and the three paths are joined by "or" statements, so we add them. Mathematically, this produces:
(0.3) + (0.7)*(0.3) + (0.7)*(0.7)*(0.3) = 0.657
Method #2 uses a powerful approach that often simplifies calculations in probability. Instead of figuring out the probability of success, we figure out the probability of failure, and then subtract that from 1. This involves the complement rule:
https://magoosh.com/gmat/2012/gmat-math- ... -question/
What would it mean for Leo
not to hear a song that he likes? It would mean that each of the three stations was playing a song he didn't like. Three individual failures, all independent, and Leo experiences all three. In words, this is
(Leo does not + A) and (Leo does not + B) and (Leo does not + C)
There's only one path that leads to this. If at any point, Leo hears a song he likes, then it is no longer part of this path and part of this scenario. On this path, all three are linked by "and", so we multiply these:
(0.7)*(0.7)*(0.7) = 0.343
That's the probability that Leo does not find a song he likes. So, the probability that he does find a song he likes is
1 - 0.343 = 0.657
Does all this make sense?
Mike
