There's a .7 chance to select a toothbrush, .2 chance to select a yogurt, and .1 chance to select a monkey treat. There is a .8 chance the toothbrush will decay, .2 chance the yogurt will decay and .5 chance the monkey treat will decay. A robot select an item and it decays. What is the probability that is was the monkey treat?
To find the probability that the selected item was the monkey treat given that it decayed, we can use Bayes' theorem.
Bayes' theorem states that the probability of an event A happening given that event B has occurred can be calculated as:
P(A|B) = (P(B|A) * P(A)) / P(B)
In this case, event A is selecting the monkey treat, and event B is that the selected item decayed.
We are given the following probabilities:
P(monkey treat) = 0.1 (chance to select a monkey treat)
P(toothbrush decays) = 0.8 (chance that a selected toothbrush decays)
P(yogurt decays) = 0.2 (chance that a selected yogurt decays)
P(monkey treat decays) = 0.5 (chance that a selected monkey treat decays)
We want to find P(monkey treat | decays), which is the probability that the selected item was the monkey treat given that it decayed.
Using Bayes' theorem, we can calculate this probability as:
P(monkey treat | decays) = (P(decays | monkey treat) * P(monkey treat)) / P(decays)
P(decays | monkey treat) is the probability that the selected item decays given that it was a monkey treat, which is given as 0.5.
P(decays) is the probability that the selected item decays. In this case, it is the probability of any item decaying multiplied by the probability of selecting each item:
P(decays) = (P(toothbrush) * P(toothbrush decays)) + (P(yogurt) * P(yogurt decays)) + (P(monkey treat) * P(monkey treat decays))
= (0.7 * 0.8) + (0.2 * 0.2) + (0.1 * 0.5)
= 0.56 + 0.04 + 0.05
= 0.65
Now, we can substitute the values into the Bayes' theorem formula:
P(monkey treat | decays) = (0.5 * 0.1) / 0.65
= 0.05 / 0.65
= 0.0769
So, the probability that the selected item was the monkey treat given that it decayed is approximately 0.0769, or 7.69%.