Let W be the number shown on a biased die. It is known that ℙ(=)=0.2 for w even, and also that the probability the die shows a prime number is 0.5. What is ℙ(=1)?
[Note: You can assume the die has six sides.]
How do i approach this question as generally i would draw a pmf table however doing this gives me more than 1 which indicates that it is incorrect, any tips?
My working out - P(w=2,4,6)=.2 P(w=2,3,5)=0.5 As 2 is common i multiplied give p(w=2)=0.10
We know that $P(W=2) = P(W=4) = P(W=6) = 0.2$, and that $0.5 = P(W\in\{2,3,5\}) = P(W=2)+ P(W=3) + P(W=5)$. This implies that $P(W=3) + P(W=5) = 0.3$.
As we also know that $\begin{align*} 1 &= P(W=1) + P(W=2) + P(W=3) + P(W=4) + P(W=5) + P(W=6)\\ &= P(W=1) + 0.2 + 0.2 + 0.2 + (P(W=3) + P(W=5))\\ &= P(W=1) + 0.6 + 0.3 \end{align*}$
we can calculate the value of $P(W=1)$.