Problem Solving

Hey guys! I just started preparing for the GMAT (less than 2 weeks). I'm an engineer so I decided to brush up on my quant skills before taking the diagnostic. I realized that I'm very weak in probability and statistics. So I'm doing a bunch of basic probability problems and I got stuck with these two problems:

1. Marvin lives in Stormwind City and works as an engineer in the city of Ironforge. In the morning, he has 3 transportation options (teleport, ride a dragon, or walk) to work, and in the evening he has the same 3 choices for his trip home. If Marvin randomly chooses his method of travel in the morning and in the evening, what is the probability that he teleports at least once per day?

2. If you flip three fair coins, what is the probability that you'll get heads on the first two flips and tails on the last flip?


I don't have the answers for these so my guess is (5/9) for the 1st question and (1/8) for the 2nd question. Can someone help with this?

Thanks!

1. Marvin lives in Stormwind City and works as an engineer in the city of Ironforge. In the morning, he has 3 transportation options (teleport, ride a dragon, or walk) to work, and in the evening he has the same 3 choices for his trip home. If Marvin randomly chooses his method of travel in the morning and in the evening, what is the probability that he teleports at least once per day.

If we were to actually calculate the answer to this one, we would have to calculate two probabilities, the probability that he teleports once and the probability that he teleports twice. So rather than do that, we can just calculate the probability of his not teleporting at all, and then subtract that from 100 percent to arrive at the probability that he teleports at least once.

The probability of his not teleporting is 2/3 in the morning and 2/3 in the evening. So the probability of his not teleporting either time is 2/3 x 2/3 = 4/9.

So the probability of his teleporting at least once is 1 - 4/9 = 5/9

2. If you flip three fair coins, what is the probability that you'll get heads on the first two flips and tails on the last flip?

This one is just the probability of getting three specific outcomes, the probabilities of each of which are 1/2.

So, yes, the probability of this is 1/2 x 1/2 x 1/2 = 1/8

@Marty - Thank you! Your approach was straightforward and definitely the most popular way of solving the problem. But I realized that this way doesn't work for me. I came across the probability tree on Khan Academy (which is where the two questions are from) and found that method to be the easiest. The method is visual which is highly helpful as I'm a very visual person when it comes to math. Maybe harder questions would require your method but I haven't gone there yet.

@Rich - Yes, I plan on taking the diagnostic on Saturday. I'm not really sure which would be a good diagnostic as there are so many CATs available online. Currently, I'm thinking Veritas should work. With regards to study material - I worked through the MGMAT quant guides the past two weeks. I brushed up on my math skills and identified my weak areas. After the diagnostic I plan on spending time with MGMAT SC and Powerscore CR guides. I do want to sign-up for a course after a month or two. I looked at EmpowerGMAT. It looks like you guys offer 1,2 and 3 months study plans but I was unable to see further details. For example, how does each plan differ from the other? What study materials do you use? Do you offer additional CATs? I have to admit that I wasn't very thorough with this research because of the time I still have to sign-up for the course. Any insight you have on this would be helpful!

@Rich - In terms of target goal and date, I'm finding it hard to pick something concrete. I want to spend 4-6 months and cross the 700 mark. It's big vague but I feel I'll be able to pick a goal somewhere along the way.

vdm, I am wondering why you didn't mention the official GMAT Prep tests as a possible choice for your first practice test. Is it that you want to save them for later?

Those tests provide the most accurate representation of the actual GMAT. So using one of them would probably give you the most accurate sense of what you need to do to hit your score goal.

Regarding what works or does not work for you, while I can understand your wanting to go with a method that just flows for you - I tend to do that also - I have noticed the degree to which people are adaptable and can learn new methods. Do something enough times and it will become second nature. That includes this test itself, and I am mentioning all this to give you added confidence that you can hit your score goal as much of people's success in scoring high on this test comes down to their own perceptions of what they are capable of.

Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer.

1. Marvin lives in Stormwind City and works as an engineer in the city of Ironforge. In the morning, he has 3 transportation options (teleport, ride a dragon, or walk) to work, and in the evening he has the same 3 choices for his trip home. If Marvin randomly chooses his method of travel in the morning and in the evening, what is the probability that he teleports at least once per day?
==> Assuming P(E) = the probability of event E = number of cases of event E / All the number of cases
All the number of cases are 3*3 = 9 = (t,t),(t,r),(t,w),(r,t),(r,r),(r,w),(w,t),(w,r),(w,w)
And the case of at least one teleports is 5 = (t,t),(t,r),(r,t),(t,w),(w,t)
Thus the answer is 5/9.

2. If you flip three fair coins, what is the probability that you'll get heads on the first two flips and tails on the last flip?
==> Since the probability is 1/2 for both Tail and Head, HHT=(1/2)(1/2)(1/2)=1/8.

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Let's show a couple approaches for each.

1::

If we do this positively, we have to consider two cases: Marvin teleports exactly once, and Marvin teleports both times. If teleports once, we have two options:

a) Teleports AM, does not teleport PM = (1/3) * (2/3) = 2/9
b) Teleports PM, does not teleport AM = (2/3) * (1/3) = 2/9

If he teleports both times, that's obviously (1/3) * (1/3) = 1/9.

Since each of these scenarios is unique, there's no overlap between them, and we can add them up safely: 2/9 + 2/9 + 1/9, or 5/9.

We can also do this negatively. If we need Marvin to teleport at least once, the only scenario we aren't interested in is the one in which he doesn't teleport at all. No teleporting at all -- Dhalsim does not approve -- has a probability of (2/3) * (2/3), or 4/9. Since this is the only scenario we DON'T want, to find the ones we DO want, we do 1 - (Scenario We Don't Want), or 1 - 4/9, which gives 5/9 again.

(Of course, the real answer should be 0: when you can ride a dragon to work, why would you ever teleport!?)

2::

We could always consider the positive scenario, which is (Heads First) * (Heads Second) * (Tails Third), or (1/2) * (1/2) * (1/2), or 1/8.

We could also think of all the ways to arrange three coins. There are two options for each coin, or 2*2*2, or 8 total. We only want ONE of these (HHT), so our probability is again 1/8.