MHB Probability Function of Z: 1/4 & 1/2 Explained

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The discussion focuses on calculating the probability function of Z, defined as the product of two variables X and Y based on the outcomes of flipping two fair coins. The probabilities are derived as follows: P(Z=0) = P(X=0) = 1/2, P(Z=1) = P(X=1) * P(Y=1) = 1/4, and P(Z=5) = P(X=1) * P(Y=5) = 1/4, with all other probabilities being zero. Confusion arises regarding why P(X=0) is not multiplied by P(Y=1) or P(Y=5), but the explanation clarifies that if X=0, Z is always zero regardless of Y's value. The final conclusion is that the calculations correctly reflect the probabilities based on the defined conditions.
oyth94
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Consider flipping two fair coins. Let X = 1 if the first coin is heads, and X = 0
if the first coin is tails. Let Y = 1 if the second coin is heads, and Y = 5 if the second
coin is tails. Let Z = XY. What is the probability function of Z?
how did you get 1/4 and 1/2 ?? and why? confused!
 
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Re: probability function

oyth94 said:
Consider flipping two fair coins. Let X = 1 if the first coin is heads, and X = 0
if the first coin is tails. Let Y = 1 if the second coin is heads, and Y = 5 if the second
coin is tails. Let Z = XY. What is the probability function of Z?
how did you get 1/4 and 1/2 ?? and why? confused!

Setting $P_{k}= P \{Z=k\}$ we have...$$P_{0}= P \{X=0\} = \frac{1}{2}$$ $$P_{1} = P \{X=1\}\ P \{Y=1 \} = \frac{1}{4}$$$$P_{5} = P \{ X=1\}\ P \{Y = 5\} = \frac{1}{4}$$ ... and for any other k is $P_{k}=0$...

Kind regards

$\chi$ $\sigma$
 
Re: probability function

chisigma said:
Setting $P_{k}= P \{Z=k\}$ we have...$$P_{0}= P \{X=0\} = \frac{1}{2}$$ $$P_{1} = P \{X=1\}\ P \{Y=1 \} = \frac{1}{4}$$$$P_{5} = P \{ X=1\}\ P \{Y = 5\} = \frac{1}{4}$$ ... and for any other k is $P_{k}=0$...

Kind regards

$\chi$ $\sigma$

How come for the "P (X=0) = 1/2" you didn't multiply by P(Y=1) or P(Y=2)? I'm not sure how you ended up with 1/2 instead of 1/4 for this one...
 
Re: probability function

oyth94 said:
How come for the "P (X=0) = 1/2" you didn't multiply by P(Y=1) or P(Y=2)? I'm not sure how you ended up with 1/2 instead of 1/4 for this one...

If X=0 then is Z = X Y = 0 no matter which is Y...

Kind regards

$\chi$ $\sigma$
 
Re: probability function

oyth94 said:
How come for the "P (X=0) = 1/2" you didn't multiply by P(Y=1) or P(Y=2)? I'm not sure how you ended up with 1/2 instead of 1/4 for this one...

If it helps, you can also think of $P_0$ as follows:
Because y=1 and y=5 are mutually exclusive events, we can state:
$$P\left(Z=0\right) = P\left(X=0\right)
\\= P\left(X=0 \wedge Y=1\right)+P\left(X=0 \wedge Y=5\right)
\\= P\left(tails \wedge heads\right)+P\left(tails \wedge tails\right)$$
The probability of the first event, as you rightly stated, is
$$\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4}$$
The probability of the second even is the same.
Adding these two together, we have
$$\frac{1}{4}+\frac{1}{4}=\frac{1}{2}$$
giving us our answer.

As $\chi \sigma$ rightly stated, we would have gotten the same answer if we had simply evaluated
$$P\left(Z=0\right) = P\left(X=0\right) = P\left(first \, coin \, is\, tails\right) = \frac12$$
 
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