Our intuition seems to be consistent with the business propaganda, and the winning rate is10/1≈ 91%. But if you stand by and watch the lucky draw, you will find that many lottery winners have won the 1 1 etc. Even other free prizes are mostly small prizes with low value. Is it a hidden organ in the box?
In fact, there is no "organ" in the box. The key problem is that the probability of winning the eleventh prize is different. Randomly draw 10 balls, and let the number of 10 balls be x, then the number of 5 balls is 10-X. X obeys hypergeometric distribution, that is, p (x = I) = ... (Sorry, the hypergeometric distribution formula cannot be written in this format, please check the hypergeometric distribution formula yourself), (. , 10)
According to the above formula, the probability of each event can be obtained as follows:
Table 1
Quantity 10 point X 0 1 2 3 4 5
Probability p (x = i) 010000050100541096010779410/238693010.
Score 50 55 60 65 70 75
Reward 13579 1 1
10 points X 6 7 8 9 10
Probability p (x = i) 01238693010779410109601005410/00005.
Score 80 85 90 95 100
Reward 18642
The probability that lottery winners need to buy products exceeds 1/ 3, and the higher the prize value, the lower the probability. In particular, the probability of winning two grand prizes is only about one in one hundred thousand. Let's study the expected income of the merchants in a lottery. After investigation, the cost of each contract is shown in the following table:
Table 2
Reward one two three four five six
Probability p (x = i) 010000005 0100005 010005410/01090.
The bonus cost is 2500 500 150 80 20 10.
Merchant income-2500-500-150-80-20-10
Reward 789 1 1
Probability p (x = I) 01077941kloc-0/2386930123869301343718.
The prize cost is 2192111201410.
Merchant income-219-211-kloc-0/2-015
The expected income of a merchant in a lottery = ∑ merchant income × corresponding probability, and the calculation result is 3 19 1 yuan. What seems to be a "free lottery" is actually a means for businesses to promote products and gain profits.