Probability Question

Xonian

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Hey guys. In my Algebra class we have been doing probability. I understand the fundamental ideas, but I have a specific question. I have attached a photo of the problem, number 37. The answer is .5, but I really would appreciate someone walking me through it. Thanks!
 
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Hey guys. In my Algebra class we have been doing probability. I understand the fundamental ideas, but I have a specific question. I have attached a photo of the problem, number 37. The answer is .5, but I really would appreciate someone walking me through it.
Toss a coin three times. Here is a table of all possible outcomes:
\(\displaystyle \begin{array}{*{20}{c}} H&H&H \\
H&H&T \\
H&T&H \\
H&T&T \\
T&H&H \\
T&H&T \\
T&T&H \\
T&T&T
\end{array}\)
There only eight possible outcomes.
How many of those rows contain exactly two \(\displaystyle T's\) or three \(\displaystyle H's\). Divide by eight.
 
pka's method is certainly the simplest way to do this. Another, applicable if you had a larger number of "tosses" where it would be difficult to right out all possible combinations is this:

The probability of any one coin coming up heads is 1/2. The probability of all three heads is (1/2)(1/2)(1/2)= 1/8.
(The denominator matching pka's 8 possible outcomes of which "HHH" was one.)

Similarly, the probability of any one coin coming up tails is also 1/2 so the probability of "heads, tails, tails" is (1/2)(1/2)(1/2)= 1/8. But there are \(\displaystyle \begin{pmatrix}3 \\ 2 \end{pmatrix}= \frac{3!}{2! 1!}= \frac{6}{2}= 3\) ways to have one head and two tails (those are pka's "HTT", "THT", and "TTH") and each has the same (1/2)(1/2)(1/2)= 1/8 probability: the probability of two tails and one head- in any order is 3/8.
Add: 1/8+ 3/8= 1/2.
 
pka's method is certainly the simplest way to do this. Another, applicable if you had a larger number of "tosses" where it would be difficult to right out all possible combinations is this:

The probability of any one coin coming up heads is 1/2. The probability of all three heads is (1/2)(1/2)(1/2)= 1/8.
(The denominator matching pka's 8 possible outcomes of which "HHH" was one.)

Similarly, the probability of any one coin coming up tails is also 1/2 so the probability of "heads, tails, tails" is (1/2)(1/2)(1/2)= 1/8. But there are \(\displaystyle \begin{pmatrix}3 \\ 2 \end{pmatrix}= \frac{3!}{2! 1!}= \frac{6}{2}= 3\) ways to have one head and two tails (those are pka's "HTT", "THT", and "TTH") and each has the same (1/2)(1/2)(1/2)= 1/8 probability: the probability of two tails and one head- in any order is 3/8.
Add: 1/8+ 3/8= 1/2.

Thank you both so much! Cheers!
 
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