Suppose that a sloth sanctuary has 14 two-toed sloths and 16 three-toed sloths,

Question

Suppose that a sloth sanctuary has 14 two-toed sloths and 16 three-toed sloths, all distinguishable.The people who manage the sanctuary want to create a calendar to raise funds. This calendar will have a picture of exactly one sloth in each of the 12 months, and a different sloth will be featured each month.

(a) What is the probability that the calendar will have pictures of exactly 10 two-toed sloths?

(b) What is the probability that the calendar will have pictures of at least 10 two-toed sloths?

(c) How many ways are there to not only choose which sloths will be on the calendar but to decide which sloth will appear on which month?

(d) How many ways are there to decide which sloth will appear on which month if the January sloth must be three-toed and the December sloth must be two-toed?

Explanation / Answer

(a) The 10 two-toed sloths can come in 14C10 ways.

The remaining 2 three-toed sloths can come in 16C2 ways.

=> Probability = 14C10 * 16C2 / 30C12

= 1001 * 120 / 86493225

= 0.001388

(b) The probability of the calendar having pictures of 11 two-toed sloths

= 14C11 * 16 / 30C12

= 364 * 16 / 86493225

= 0.00007

The probability of the calendar having pictures of 12 two-toed sloths

= 14C12

= 91 / 86493225

= 0.000001

=> Probability of having atleast 10 two-toed sloths

= 0.001388 + 0.00007 + 0.000001

= 0.001459

(c) The first month can have pictures of any of the 30 sloths.

The second can have pictures of 29 sloths

and so on.

=> Total number of ways to choose sloths and order

= 30*29*...19 or 30P12

(d) The number of options for January = 16

Number of options for December = 14

For the remaining 10 months, the number of options = 28C10

=> Total number of ways = 16 * 14 * 28C10

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