I have the following real life problem to solve, about the use of a scarce resource. I'll explain it with an example, as I am a physician and not a mathematician, but I hope to find help. Suppose we have 4 urns (A B C D) containing red and blue marbles
- The urn A contains 2 red marbles out of 100 marbles (2 red and 98 blue)
- The urn B contains 5 red marbles out of 100 marbles (5 red and 95 blue)
- The urn C contains 12 red marbles out of 100 marbles (12 red and 88 blue)
- The urn D contains 30 red marbles out of 100 marbles (30 red and 70 blue)
If I randomly select 5 marbles from the urn A, 2 marbles from the urn B, 4 marbles from the urn C, and 1 from the urn D (all in a raw, with replacement)
what is the probability of the event: getting a red marble?
(I don’t know what formula or model should I use. I think I could use binomial distribution if only one urn was present I think multinomial distribution is not for this case. Any suggestion?).Thank you
Consider this: the probability of getting a red marble is $1-P^*$, where $P^*$ is the probability of not getting a red marble. When you draw $5$ times from urn $A$, the probability of getting no red marbles is $$P^*_A=\frac{98}{100}*\frac{97}{99}*\frac{96}{98}*\frac{95}{97}*\frac{94}{96}$$ from urn $B$, the probability of getting no reds is $$P^*_B=\frac{95}{100}*\frac{94}{99}$$ from urn $C$, it is $$P^*_C=\frac{88}{100}*\frac{87}{99}*\frac{86}{98}*\frac{85}{97}$$ and from urn $D$, it is $$P^*_D=\frac{70}{100}$$ Then the probability of getting no reds is $$P^*=P^*_AP^*_BP^*_CP^*_D$$ and the probability of getting at least one red is $$1-P^*$$