# Thread: binomial distribution dice and coins

1. ## binomial distribution dice and coins

Hey, so I have 2 questions for now that I really need some help on. The first is...
1. A person rolls a pair of fair dice. What is the probability that this person will roll no 7's on three rolls

2. If five coins are tossed simultaneously, what is the probability of obtaining exactly two tails? Is the result the same if one coin is tossed five times?

~If it is at all possible to give really detailed step by step guide to these questions it would be really helpful.

Thanks
~G

2. Originally Posted by the-G
Hey, so I have 2 questions for now that I really need some help on. The first is...
1. A person rolls a pair of fair dice. What is the probability that this person will roll no 7's on three rolls
see the diagram i have in post #2 here

from that diagram, you can find the probability of getting a sum of 7 on any one roll. take that as your probability of success, call it p. let q = 1 - p be the probability of failure. then your desired probability, given by the Binomial distribution would be (if we let X be the number of 7's we get):

$P(X = k) = {3 \choose k} p^kq^{3 - k}$

here you want k = 0

you could also use logic and go around the binomial distribution (though you would be using it implicitly), by finding the probability of not getting a 7, again by the diagram, and cubing the probability, since we want the intersection of the event that we get no 7 with itself, three times

3. Originally Posted by the-G

2. If five coins are tossed simultaneously, what is the probability of obtaining exactly two tails?
the outcome of the 5 coins will be independent of each other and we can classify the outcomes as "success" or "failure", thus we can use the binomial distribution here.

we want in 5 trials two "successes" of obtaining heads, thus, the probability is:

$P(2) = {5 \choose 2}p^{2}q^{5 - 2}$

what do you think p and q are here?

Is the result the same if one coin is tossed five times?
I would say yes. but i recall there being a discussion on the forum recently as to whether the contrary may be true. i don't recall the outcome of the discussion. i shall try to search for it and tell you if i find it.

you may also want to do a search for "binomial" or "binomial distribution" or "binomial expansion" or "regents" on this forum, and you should find questions like this, where a lot of explanation is given

here is the post i mentioned. it seems there was not closure, but you can read it to get some ideas. a textbook was quoted, so you can perhaps find a similar section in your text

4. ## ?

I still find it really confusing.

For the second question I thought that q and p would be 3/5 and 2/5, respectively. but it doesn't work out.

For the first question I don't understand. why would X be the number of successes when we are looking for the number of time we wouldn't roll a 7, wouldn't that number be the number of successes?

thanks for taking the time to help me.

5. Originally Posted by the-G
I still find it really confusing.

For the second question I thought that q and p would be 3/5 and 2/5, respectively. but it doesn't work out.
when flipping a coin, what is the probability of getting tails? that is your p

For the first question I don't understand. why would X be the number of successes when we are looking for the number of time we wouldn't roll a 7, wouldn't that number be the number of successes?
i said X was the number of 7's. it is the number of successes we want, which is 0

6. Note that a pair of dice is rolled. Therefore it is of course possible to role a 7.
The probability is $\frac {6} {36}$.
So the probability of no 7’s in three rolls is $\left(\frac {30} {36} \right)^3$ .