"Billy goes to his grandparents house to help out with chores, he's always running late though.
The probability he is late when he walks is 30% The probability he is late when he runs is 15% The probability he is late when he bikes is 5% The probability he walks is 0.2 The probability he runs is 0.1 The probability he bikes is 0.7
What is the probability he will be late?
Billy ended up being late once again, what is the probability he walked, what is the probability he ran?"
I am stuck on how to set this up using Bayes theorem Would it be: 0.2 * 0.1 * 0.7 for being late Then, 0.3 * 0.2 / 0.3 * 0.2 + 0.2 (late that he didn't walk, which is 0.15 + 0.5) * 0.8 ( this is 0.7 + 0.1, where he ran or biked, not walked)
The probability to be late is
$$\mathbb{P}[L]=0.3\times0.2+0.15\times0.1+0.05\times0.7=0.11$$
Thus, simply dividing, you get
$$\mathbb{P}[W|L]=\frac{0.3\times0.2}{0.11}\approx 54.55\%$$
$$\mathbb{P}[R|L]=\frac{0.15\times0.1}{0.11}\approx 13.64\%$$