Question

The probability that you roll a 5 on a six-sided die is start fraction one over six end fraction . The probability that you flip a coin that lands on heads is start fraction one over two end fraction. The probability that you roll a 5 on a six-sided die and you flip a coin that lands on heads is start fraction one over 12 end fraction. What is the probability of flipping a coin and it landing on heads, given that you rolled a 5 on a six-sided die? Are these two events independent?

Answer 1

I am not sure about the probability but I cant tell you these events are independent because u don't need to roll a five for a coin to land on heads

Answer 2

Let:

F - 5 on a six-sided die,
H - coin lands on heads

We also know that:

$P(F)=\dfrac{1}{6}\qquad\qquad P(H)=\dfrac{1}{2}\qquad\qquad P(F\cap H)=\dfrac{1}{12}$

Calculate:

$P(H|F)=\dfrac{P(F\cap H)}{P(F)}=\dfrac{\frac{1}{12}}{\frac{1}{6}}=\dfrac{1\cdot6}{12\cdot1}=\dfrac{6}{12}=\boxed{\dfrac{1}{2}}$

In this case events are independent when:

$P(H|F)=P(H)$

Both probabilities equals $\dfrac{1}{2}$, so H and F are independent.