Probability questions involving percentages

Question

Lola is obsessed by the colour of her hair. On any given day there is an 80% chance she will change the colour of her hair for the next day. Her hair is blond 40% of the time, brown 30% , red 20% and purple for the remainder. Given Lola has red hair on Friday, what is the probability that

1. Tomorrow her hair is brown ?
2. Her hair is not red on Saturday and Sunday AND her hair is a different colour on Saturday and Sunday.

I have trouble with this probability question, even though I read the answers but I still don't understand their solution. In the solutions for Q1, they simply just wrote $0.8 \times \frac{3}{8}=0.3$, I understand where the $0.8$ came from since there is a 80% chance that she will change her hair. But where did the $\frac{3}{8}$ come from? Thanks.

Answer 1

I think an interesting way to think of your question one is to imagine Lola without hair on Friday. On Saturday we know there is an 80% chance of changing so the first (0.8) is clear. Now if she doesn't have hair, there is a 30% chance of getting brown hair so (0.8)*(0.3) or $\frac{8}{10}$ * $\frac{3}{10}$ which is likely what you originally thought.

However, we know Lola does have hair and red hair at that. Red hair has a 20% chance of occurring so our formula looks like

$\frac{8}{10}$ *$\frac{30}{100-20}$ similarity, $\frac{8}{10}$ *$\frac{30}{40+30+10}$

The 100-20 is really meaning that there was a 100% chance but we've taken off 20% from the total and the 40+30+10 is a sum of all the other probabilities!

Answer 2

If the color is red, then changing means that it will be either blond, brown, or purple. It cannot be red. So the chance to be brown is $$\frac{30}{40+30+10}=\frac 38$$

Answer 3

They are assuming that if she decides to change the color, the color she changes to is in proportion to the long term averages of the other three colors. That is one way to achieve the long term averages but not the only way. Another way she could decide hair color is to alternate between (blond or purple) and (brown or red). Each time she changes, she changes to one of the other pair in the proportion of the long term odds. In that case, if we know her hair was red on Friday, she has $20\%$ chance to have red hair on Saturday, $\frac {40}{40+10}\cdot 80\%=64\%$ chance to have blonde, $16\%$ chance to have purple, and $0\%$ chance to have brown. This way will also satisfy the long term odds. There are many others.