Say, I had a button that had a 50% chance to make me go bald and a 50% chance to let me press another button, called Button Love. Button Love, when pressed, gives me a 10% chance to find true love. I can only press the first button if I have at least 4 inches of hair, which requires 4 months for me to grow out. On average, how many months will I spend before I find true love.

Thing to notice; If I don't go bald on my first press of the first button, I can press it again and again until i eventually do go bald.

1. **Going bald on the first press of Button 1:** This happens with a probability of 0.5. In this case, you won't be able to press Button Love, so the time spent is 0 months.

2. **Not going bald on the first press of Button 1:** This also happens with a probability of 0.5. If you don't go bald, you'll keep pressing Button 1 until you eventually do. Since it takes 4 months to grow back 4 inches of hair, you'll spend, on average, 2 months pressing Button 1 before going bald.

Now, for each time you press Button 1 without going bald, you have a 50% chance of finding true love by pressing Button Love, which has a 10% success rate.

So, the expected number of months spent pressing Button Love each time is [tex](2 \text{ months} \times 0.5 \text{ chance of not going bald} \times 0.1 \text{ chance of finding true love}) which equals (0.1 \text{ months})[/tex]

Therefore, on average, you'll spend [tex](2 \text{ months})[/tex]pressing Button 1 each time and [tex](0.1 \text{ months})[/tex] pressing Button Love. So, the total time spent before finding true love is [tex](2.1 \text{ months})[/tex]