NBA DRAFT Ping Pong Balls Explanation: As a Boston Celtics fan, my biggest dream came true on the last night of the season. The Celtics lost at home to the surging Wizards and the Jazz won on the road against the Timberwolves. This and some earlier favourable results catapulted (consigned?) the Celtics to the joint 4th worst record in the NBA, and with it, a supposed 10.4% chance of nabbing the top pick in this year’s draft. That is all good you say, but how do you arrive at this number? It is not very simple, but is discernable. Let us start off with a simple example…

You have 3 balls, labelled 1,2 and 3. Let us say that you pick these balls randomly out of a box and arrange it in the order in which you picked it. How many combinations could there be? Let us have a go..

List 1

1,2,3

2,3,1

3,1,2

3,2,1

2,1,3

1,3,2

There is a very simple way to calculate the number of combinations that arise from such random selections. For 3 balls, it is simply 3 *2 *1 = 6. For 4 balls, it would be 4 * 3 *2 * 1 = 24. For 5 it would be  5 * 4 * 3 * 2* 1 = 120. In essence, for n balls, the number of combinations would be n * (n-1) * (n-2) *…* 1.

Getting back to our example, we have 6 combinations that result from your selections from the box. Now, the NBA Draft Lottery uses exactly the same system. Except, that it does not care for the order of the balls. What do I mean by that? Let us go back to our example. We have 2 sets of combinations with 2 in the centre: 1,2,3 and 3,2,1. In the lottery, both these combinations are considered the same. Yes, the order in which the numbers are picked does not matter. So, what started off as 6 combinations have been whittled down to 3. 1,2,3 ,  2,3,1  and 3,1,2. Or any 3 combinations with 1,2 and 3 respectively in the middle once each.

Now, in the Draft Lottery, there are 14 ping pong balls painted 1-14, placed in the lottery machine and 4 are selected at random. The 4 selected could be 1,2,3,4 or 4,2,3,1 or any of several combinations. But remember, order does not matter. So this question essentially becomes a simple matter of choosing 4 balls from 14 without really caring about the order.

There is a very simple way to find this out. If in our old example, we had to select 2 balls from 3 and if the order does not matter, how many combinations would we have? Let us check – (2,3),(3,2),(1,3),(3,1)(1,2)(2,1). But the order does not matter so (1,2)=(2,1) and so on. So, we are left with 3 combinations.

Let us take another example. Say we had 4 balls and we had to choose 3 at a time. What would we have then? 4 combinations as the order simply does not matter. Turns out there is a very easy way to calculate the number of combinations in this case. It is simply the n * (n-1) * (n-2) *…* 1 we had before divided by the product of (n-r)*(n-r-1)*(n-r-2)..*1 and r*(r-1)*..1 where r is the number of balls picked at a time. So for our 4 ball example where we have to pick 3 at a time, we would have : (4*3*2*1)/( (4-3)*(3*2*1)) = 4.

Now to the Draft Lottery.

There are 14 ping pong balls from which 4 will be selected. So the number of combinations will be (14*13*12*11*10*9…*2*1)/(((14-4)*((14-4)-1)* ((14-4)-2)*..*1)*(4*3*2*1)) = 1001 combinations. Out of this, the combination of 11,12,13,14 is discarded. So, there are 1000 combinations in total.

Out of these 1000 combinations, 250 combinations are randomly assigned to the team with the worst record. 199 to the team with the second worst record, 156 for the team with the third worst and so on. Now if there is a tie, the ping pong balls are divided between the teams. For instance the Jazz and the Celtics own the joint fourth worst records. So the total number of ping pong balls between the 4th and 5th worst records is averaged between the two. The fourth worst team gets 119 combinations and the fifth, 88. So the Celtics and Jazz will average this out = (119+88)/2 = 103.5. Now, as there cannot be half a ball, a toss will decide which team will get 104 and which gets 103.

NBA Draft Ping Pong Balls Explanation: So, when the NBA representative picks out his 4 balls from the lottery machine, the 4 ball combinations could be with any of the lottery bound teams. The worst team has got a better probability of matching the choice. For instance, Milwaukee this year will have 250 of the 1000 combinations so the probability of them having been assigned the same 4 balls as what the NBA representative picks out is 250/1000=25%. That is a 1 in 4 chance of nabbing the top pick. The Jazz and Celtics will approximately have 103/1000 = 10.3% chance of doing so. If the Jazz had lost on Wednesday they would have had 119 combinations (11.9%)and the Celtics 88(8.8%). No wonder the Jazz fans were fuming at their team’s inconsequential double OT victory which just reduced their team’s chances of a top pick by about 1.6%! Not that I am complaining. Hope you liked our explanation on ping pong balls NBA.

The Celtics loss coupled with Jazz’s win increased the Greens’ chances of a top pick by 1.5%. Now, that is the kind of silver lining I was looking for in an otherwise forgettable season for the Celtics!