- **Precalculus Mathematics Homework**
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- - **50 coin tosses, probability of 25 heads vs 3 heads**
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50 coin tosses, probability of 25 heads vs 3 heads1. The problem statement, all variables and given/known dataToss a coin fifty times. What is the probability of throwing 25 heads, and the probability of throwing 3 heads? 2. Relevant equationsP = 0.5 ^ n, n = total tosses other equations??? 3. The attempt at a solutionthe probability for each should be same, as in 0.5^50, for each, should it not? when i asked this question in class, my statistics teacher gave me a weird look and moved on with the class. am i missing a fundamental point in logic or something more complex? |

Re: 50 coin tosses, probability of 25 heads vs 3 headsH=head
P(H)=0.5 P(HH)=P(H)*P(H) (this is because they are independent events. So that getting a head on the first try does not influence the outcome of the second) |

Re: 50 coin tosses, probability of 25 heads vs 3 headsOk, so you throw the coin 50 times. That is clear.
My first question, must that 25 heads be in row? Also, do you know what probability is? P=m/n , m - the number of possibilities that might occur, n - total number of possibilities Also [itex]0 \leq P \leq 1[/itex]. In addition, just imagine if you throw the coin once, what is the probability that head or tail will be thrown? What is the probability for 50 times? |

Re: 50 coin tosses, probability of 25 heads vs 3 headsQuote:
specific order- say 25 heads followed by 25 tails or alternating heads and tails. There are [tex]\left(\begin{array}{c}50 \\ 25 \end{array}\right)[/tex] ways of ordering 25 heads and tails so the probability of getting 25 heads and 25 tails in any order is [tex]\left(\begin{array}{c}50 \\ 25 \end{array}\right)(.5)^{50}[/tex]. Similarly, there are [tex]\left(\begin{array}{c}50 \\ 3 \end{array}\right)[/tex] ways of arranging 3 heads and 47 tails so the probability of flipping 3 heads and 47 tails in any order is [tex]\left(\begin{array}{c}50 \\ 3 \end{array}\right)(.5)^{50}[/tex] |

Re: 50 coin tosses, probability of 25 heads vs 3 headsQuote:
-are those 25 heads thrown in a row? -are there 25 heads and are they ordered?at leastI believe its the first one because the second one is too complicated to solve. In this case combinations work as same as permutations with repetition. In cases like this one, I always take smaller values and find the pattern :smile: For example, lets imagine that instead of 50 there are 5 throws. We need 3 heads (not ordered) in each 5 throws. So, the string will look like HHHTT , H - head ; T - tail. Instead of permutations, we need permutations with repetition, so: [tex]P_{(3,2)}5=\frac{5!}{2!3!}=10[/tex] And the probability would be 10/2 ^{5}.Do the same for the original problem. |

Re: 50 coin tosses, probability of 25 heads vs 3 headsYou are reading too much into the problem, njama. These problems are obviously asking about combinations, not permutations. If the questions were to find the probability of getting at least 25 heads (or 3) out of 50, they would have said just that.
Looking for patterns works for a small number of objects. This technique breaks down quickly as n increases. |

Re: 50 coin tosses, probability of 25 heads vs 3 headsQuote:
m - permutations with repetition n - variations with repetition. P=m/n=P _{(25,25)}50/2^{50}The technique of doing the same task with smaller amounts, let you analyze the problem and give you an idea how to solve it. Nothing more. :smile: |

Re: 50 coin tosses, probability of 25 heads vs 3 headsMinor point of clarification: A permutation with repetition of objects of exactly two kinds (like heads and tails)
is a combination. They are the same thing.--Elucidus |

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