# Three trees: What's the chance of having at least one male and one female?

I have 3 trees.

• These particular trees are dioecious (male or female).
• I don't know the gender of any of the trees.
• The chance of a tree being male or female is 50/50.
• I need at least one male and one female for successful pollination to occur.

What are the chances that I have at least one male and one female tree?

• Another Hint: Binomial Distribution – Graham Kemp Apr 30 '18 at 22:48

HINT: Let $M$ denotes male tree, $F$ — female tree. The probability is $$1-P(MMM)-P(FFF).$$ But, from independency, $P(MMM)=P(M)P(M)P(M)=(1/2)^3$.

BTW: What are female trees? Have they a hollow? ;-)

• Some species are dioecious; each tree having either male or female flowers. Eg: Willow, Juniper, Teak... – Graham Kemp Apr 30 '18 at 22:57
• @GrahamKemp Thank you. I knew such other plants, but not trees. – Przemysław Scherwentke Apr 30 '18 at 23:02
• @Wilson Please see my extended hint. Can you finish now? – Przemysław Scherwentke May 1 '18 at 16:49
• @Wilson Oh, sorry! Misprint. Corrected. So $1-(1/8)-(1/8)=3/4$. – Przemysław Scherwentke May 1 '18 at 17:09

This is a binomial distribution with $n=3, p=\frac12$. Let $X$ shows the number of male trees, then: $$P(X=1)+P(X=2)={3 \choose 1}\left(\frac12\right)^1\left(\frac12\right)^{3-1}+{3 \choose 2}\left(\frac12\right)^2\left(\frac12\right)^{3-2}=\\ 3\cdot \frac12\cdot \frac14+3\cdot \frac14\cdot \frac12=3\cdot \frac14=\frac34.$$

Here is my layman's interpretation of @PrzemysławScherwentke's answer:

1 - (1/2 * 1/2 * 1/2) - (1/2 * 1/2 * 1/2) = .75

Here are all of the possible combinations that I can come up with:

{M,M,M}
{M,M,F}
{M,F,M}
{M,F,F}
{F,M,M}
{F,M,F}
{F,F,M}
{F,F,F}


There are a total of 8 possible combinations. Of the 8, only 6 of them have at least one male and one female.

Therefore, I think the probability of getting at least one male and one female is 6 / 8 = 75%.