<s>[INST] Probability of picking a certain fruit from a tree A tree has 20 fruits. 15 of which have no seeds and the rest do have seeds. A bird eats 5 of these fruits picked at random.
a) If i pick one fruit from whats left on the tree whats the probability it has seeds
b) Given that this one fruit i pick has seeds whats the probability that the bird had consumed at least one with seeds.
What i have so far:
So before the bird eats anything the probability of a fruit with no seeds is 15/20=0.75 and the probability of fruit with seeds is 5/20=0.25. 
For part a is the probability that the fruit i pick(i.e. 6th fruit picked) has seeds is given by (15/20)* (14/19)* (13/18)* (12/17) *(11/16) *(5/15)=0.065
Am i on the right track with part a? As for part b im not sure where to begin.
 [/INST] Since there bird eats only five fruits, it's relatively easy to consider all possible cases. Let's write


*

*$P_0 = P(\text{bird eats zero fruits with seeds}) = \frac{15}{20} \frac{14}{19} \frac{13}{18} \frac{12}{17} \frac{11}{16} = \frac{1001}{15504}$

*$P_1 = P(\text{bird eats exactly one fruit with seeds}) =$ $5\choose 1$$\frac{5}{20} \frac{14}{19} \frac{13}{18}\frac{12}{17} \frac{11}{16} = \frac{5005}{15504}$

*$P_2 = P(\text{bird eats exactly two fruits with seeds}) =$ $5\choose 2$$\frac{5}{20} \frac{4}{19} \frac{13}{18}\frac{12}{17} \frac{11}{16} = \frac{715}{3876}$

*$P_3 = P(\text{bird eats exactly three fruits with seeds}) =$ $5\choose 3$$\frac{5}{20} \frac{4}{19} \frac{3}{18}\frac{12}{17} \frac{11}{16} = \frac{55}{1292}$

*$P_4 = P(\text{bird eats exactly four fruits with seeds}) =$ $5\choose 4$$\frac{5}{20} \frac{4}{19} \frac{3}{18}\frac{2}{17} \frac{11}{16} = \frac{55}{15504}$

*$P_5 = P(\text{bird eats exactly five fruits with seeds}) =$ $5\choose 5$$\frac{5}{20} \frac{4}{19} \frac{3}{18}\frac{2}{17} \frac{1}{16} = \frac{1}{15504}$


So for part a, you can consider the five cases separately.


*

*If the bird ate zero fruits with seeds, there are $5$ out of $15$ fruits with seeds left. So you have to multiply $\frac{1001}{15504}\frac{5}{15}$

*If the bird ate one fruit with seeds, there are $4$ out of $15$ fruits left with seeds. So you have to multiply $\frac{5005}{15504} \frac{4}{15}$


etc ...
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