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What the probability of getting 2 consecutive heads in a total of N tosses (I found this one pretty hard and I didn't figure out the right answer at the time.) Do you mean Prob(at least 1 observation of 2 consecutive heads appears) ?? Then since the subsets can be: 1. all tails 2. 1 head 1 tail 1 head 1 tail .... 3. at least 2 heads consecutively appear once So Pr(3) = 1- Pr(1) - Pr(2) = 1- (1/2)^N - (1/2)^N = 1 - power(1/2,N-1) I think the answer should be this: Since the probability of getting two heads in a row and getting two tails in a row is the same, we only need to figure out the probability of the total. The remaining event is: 1 head 1 tail......due to different order, events should be 2 Therefore the probability=0.5(1-2(1/2)^N)=0.5(1-(1/2)^(N-1)) Only solution I could think of, which took quite some time, was to actually start by drawing a tree of possible outcomes. From there, I could see only three different states at each node. Either I am in a state similar to my initial state (when I toss a tail), I can also be in tilt state (I have just flipped heads after flipping a tail) or I am in the done state (I have just flipped heads after having flipped heads before). I consider that someone would only continue flipping if he/she has not flipped two heads in a row. That is, is not in the done state. That is, for any toss level N, it wrong to assume that number of nodes is equal to 2^N. It will be strictly smaller as of level N = 3. Anyway, after 3-4 levels drawn, one should see a pattern, that is, the number of tails, or initial state nodes (is), at a level N is equal to the number of is's I had at level N-1 + the number of is's I had at level N-2. We can thus write: is(N) = is(N-1) + is(N-2). we then realize that at every level N, the number of tilt nodes (t) is equal to the number of initial states from the previous level. That is: t(N) = is(N-1) That makes sense, because on the level N-1, all initial states output one tail (becoming a new initial state at level N) and a heads (becoming a tilt state at level N). the other states of level N-1 are either done (so no continuation) or tilt. The possible outcomes of a tilt state are either done or initial state (if I flip heads again, I am done, so I can't go from tilt to tilt.) One then realize that following the same logic, the number of done (d) nodes for a level N is given by: d(N) = t(N-1) = is(N-2) that is, at every level (as of the second level, or two tosses), I will have a number of done outcomes equal to the number of initial states to levels (or tosses) ago. So the general recursive formula to get the number of done nodes for a certain level really is: d(N) = is(N-2) = is(N-3) + is(N-4) We note, by definition: is(N) = 0, is N<0 is(N) = 1, if N = 0, or 1 is(N) = is(N-1) + is(N-2) otherwise. Furthermore, the probability of being at any node of a level N is simply (1/2)^N. So if I want to find the probability that I observed at least one occurrence of 2 heads in a row. I simply add the number of done nodes from level 1 to N multiplying them by the probability of them occurring at each level. That is: P(hh after N tosses) = sum(from i = 1 to N) (d(i).(1/2)^i). E.g. for N = 2, I get: d(1).1/2 + d(2).(1/4) = s(-1).1/2 + s(0).(1/4) = 0 + 1/4. We know that this is true. you can test it for a larger number of tosses and see it is true. as N goes to infinity, the sum will converge to 1, as I have theoretically no chance of never tossing 2 heads in a row if I throw a coin an infinity number of times. I haven't found a solution that didn't use recursions. There probably is one, I'd be interested in seeing one. Show More Responses One or more comments have been removed. |
Roll a fair dice, if it's a prime number, you pay me that number times $100, if it's not a prime number, I pay you that number times $100. Consider 1 as a prime number. How much willing to pay to play this game. |
What is the expected value to you of the following game: You and another person have 3 coins each. You both flip all of your coins and if your three coins show the same number of tails as the other person, you pay them $2, otherwise they pay you $1. |
If you have a deck of cards split into 4 piles and was offered 1:1 odds to draw a face card (J Q K A) from at least one of the piles, would you take the game? Why or why not? |
What is 29^2? |
How many degrees separate the hands on an analog clock at 3:15? |
What are odds of getting EXACTLY two (2) heads if you flip a fair coin 10 times? |
When is an american call out of the money |
If you roll a dice and and receive a dollar for whatever number you role (i.e. $1 for rolling a 1, $2 for a 2 etc.), what is the expected value of rolling this dice One or more comments have been removed. |
Would you take 10 to 1 odds that a coin will land on heads 4 times in a row? |
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