[Math] Probability that 5 out of 7 bulbs will produce white flowers

probabilityprobability distributionsstatistics

If you have a bag with 25 tulip bulbs that will grow into white, yellow or red flowers. You want to plant 7 bulbs.

  1. Each bulb in the bag, independently of the others, grows into a white tulip with probability 0.6, a yellow tulip with probability 0.1, or a red tulip with probability 0.3. What is the probability that out of these 7 bulbs exactly 5 will produce white flowers?

  2. It is known that the bag contains exactly 9 bulbs that will produce white tulips, 6 bulbs that will produce yellow tulips, and 10 bulbs that will produce red tulips. (Still it is not known which color will be produced by a particular bulb. The bulbs are mixed in the bag.) What is now the probability that out of these 7 bulbs exactly 5 will produce white flowers?

For 1. I used the Hypergeometric distribution. I'm not sure if this is right though?

$P(X=k)=\frac{\binom{M}{k}\binom{N-M}{n-k}}{\binom{N}{n}}$

since a white tulip has a probability of $0.6$, M becomes $0.6*25=15$, $N= 25$,
$k=5$, $n=7$

$P(X=5)=\frac{\binom{15}{5}\binom{10}{2}}{\binom{25}{7}}$

Can 2. be solved using the same calculation? I assume not. Hope someone can help me understand the difference between 1. and 2.

Thanks!

Best Answer

  1. Keyword: Independently.

The count of White flowers is the result of a series of independent Bernoulli trials.   That is that it has a binomial distribution.

  1. Now here we have a hypergeometric distribution.   It's the count of favoured results in a sample size $n=7$ selected without replacement from a population of size $N=25$ containing $M=9$ favoured items.
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