# Probability and De Morgan's laws question.

1. Mar 6, 2013

### peripatein

Hi,
1. The problem statement, all variables and given/known data
A coin is tossed three times. I am asked for the probability that EITHER the first toss yields heads, OR the second toss yields tails, OR the third yields heads. I am expected to use De Morgan's Laws but am not sure how to define the events the themselves. I'd appreciate some guidance.

2. Relevant equations

3. The attempt at a solution
I have found the reuested probability using the following table
First toss: HTT
Second toss: HTH
Third toss: TTH
to be 3/8.
But how may I show that rigorously? Mind you, we haven't dealt with multiplication of probabilities (so (1/2)3 for each case, let's say, would not be legit). We have hitherto only dealt with De Morgan's Laws, P(AUB)=P(A)+P(B)-P(A and B), and P(AUBUC)=P(A)+P(B)+P(C)-P(A and B) - P(B and C) - P(A and C) + P(A and B and C).
As mentioned, I'd appreciate your assistance.

2. Mar 6, 2013

### haruspex

Suppose A, B, C are the events that, respectively, 1st, 2nd 3rd tosses produce a H. Write A' etc. for tail event. Can you write the expression for the desired compound event?

3. Mar 6, 2013

### peripatein

Hi haruspex,
Following your notation, I believe the probability should hence be P(A U B' U C) = P(A) + P(B') + P(C) - P(A and B') - P(B' and C) - P(A and C) + P(A and B' and C), but this yields 7/8!

4. Mar 6, 2013

### peripatein

Whereas the correct answer should be 3/8, should it not?

5. Mar 6, 2013

### Ray Vickson

If you have not yet had 'multiplication of probabilities', what are you expected to do? Somehow, you need to use the fact that the outcome of the first toss does not affect the chances of the outcomes on the second toss, and so forth. You *could* argue that the probability of the event you are given is the same as the probability of H on toss 1 OR H on toss 2 OR H on toss 3. It might be easier to see what is the probability of this latter event.

Last edited: Mar 6, 2013
6. Mar 6, 2013

### peripatein

First of all, could someone please clarify whether the answer should be 3/8 or 7/8?

7. Mar 6, 2013

### Ray Vickson

First you need to explain how you got your answer, since you claim you cannot use the "multiplication of probabilities".

Anyway, I don't think we are allowed to tell you exactly what the answer is; we are only allowed to supply some hints.

8. Mar 6, 2013

### peripatein

Well, 3/8 was derived using multiplication, and as delineated above in my first post to this thread, which is why I decided to ask for assistance here as I am unable to arrive at that answer without resorting to multiplication and using De Morgan's Laws.
7/8 was arrived at using the formula I noted above, and presumably following haruspex's rationale, again using multiplication.
Would you be more inclined to assist me now?

9. Mar 6, 2013

### haruspex

- would HHH be a 'success'?
- what combinations of toss outcomes would not be a success?

10. Mar 6, 2013

### Ray Vickson

I tried to: I said:
"You *could* argue that the probability of the event you are given is the same as the probability of H on toss 1 OR H on toss 2 OR H on toss 3. It might be easier to see what is the probability of this latter event."

11. Mar 6, 2013

### peripatein

There is more than one combination which would not be a success, so probability (I think) should be 3/8. For instance, both HHH and TTH would not be a success. Am I right?

12. Mar 6, 2013

### peripatein

The only successful combinations are:
HHT, TTT, and THH.
Unless I am mistaken.

13. Mar 6, 2013

### haruspex

You are mistaken. Success was defined as "EITHER the first toss yields heads, OR ... OR ..." . A sequence of HHH satisfies "the first toss yields heads". No need to check any further.

14. Mar 6, 2013

### peripatein

But it is "OR"! Hence, if you first get heads then the third cannot also be heads. I believe I am right. Question is how to demonstrate that using De Morgan's Laws.

15. Mar 6, 2013

### Ray Vickson

NO, no, no! OR mean either/or, so *both* can happen. Please jetisson right now any ideas you may have that OR means 'one or the other but not both'; if we want that, we say it explicitly, or else say something like "XOR".

Don't take my word for it; look it up on-line or in a textbook.

16. Mar 6, 2013

### peripatein

Even if the word "OR" in the question itself was put in bold, so as to, apparently, imply "XOR"?

17. Mar 6, 2013

### Curious3141

When we use "OR" in the English language, the most common implication is as "XOR", e.g.

"You can have coffee OR tea" generally excludes the option of having both beverages.

When you want to imply the Boolean logic understanding of "OR", the usual convention is to state "OR both": e.g. "You can have coffee OR tea OR both".

Mathematical word problems (like this one) are written in English, so you still need to interpret them accordingly.

(EDIT: However, the correct interpretation of this (as with most other probability problems) involves a Boolean OR)

Draw a tree diagram to represent events here (see attachment).

The circles in red are the only events that you're interested in. Remember to apply the laws of conditional probability as you go through the tree (this involves multiplication because of the implicit "AND" condition).

That diagram pretty much represents the exact understanding a lay speaker of English would have of events. EITHER the first toss is heads OR ... (and at this point, you consider only events stemming from the first toss coming up tails), and so forth.

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Last edited: Mar 7, 2013
18. Mar 6, 2013

### Ray Vickson

Bold would not do it, but maybe---just maybe---the word "either" might. However, I have seen hundreds of questions like this in which one is asked for the probability of either this or that or something else, and "ordinary" OR was meant. That is standard language.

However, not being clairvoyant, I cannot say with certainty what the questioner meant.

19. Mar 7, 2013

### Curious3141

As I wrote, it doesn't actually matter for this problem.

EDIT: It actually matters, and this is using a Boolean "OR" as is the usual convention.

Last edited: Mar 7, 2013
20. Mar 7, 2013

### Ray Vickson

Actually, it matters a lot, because in PROBABILITY (rather than standard English) the usual interpretation of "OR" is the inclusive one---either one or the other or both.

Of course, some types of dichotomies are automatically XOR, such as getting either a head or tail in a single toss, or a subject being either male of female, or whatever. However, when it comes to "compound" events that is often not the case. Let me repeat: I have seen hundreds of questions of this type in which the word "OR" means the inclusive one---no lie. In some other areas of discourse, maybe things are different, but not usually in probability.

However, perhaps the OP knows what his instructor or textbook usually means by OR, and he can use that.