Probability question

An ice cream company makes ice cream bars on sticks which sell for 10 cents. Suppose that the company puts a star on every 50th stick; anyone who buys a bar with a starred stick gets a free ice cream bar. If you decide to buy ice cream bars until you get a free one, how much would you expect to spend before getting a free bar?

Then consider it as ice candy instead of ice cream.

Since you say every 50th has a star the average drops due to the incease in odds of obtaining the star stick after each consecutive purchase.

This leaves out the possibility that there are other customers competing for the star.

Krumple wrote:

Since you say every 50th has a star the average drops due to the incease in odds of obtaining the star stick after each consecutive purchase.

Incorrect.

Krumple wrote:

This leaves out the possibility that there are other customers competing for the star.

Irrelevant.

The probability can not be the same if its garanteed by the 50th purchase. Which is why I am asking.

Why assume? So you are saying "you" are the only one buying? If there are others then it changes the maths and the outcome. So yes its relevant.

Krumple wrote:

The probability can not be the same if its garanteed by the 50th purchase. Which is why I am asking.

You're assuming that there are only 50 ice cream bars. That's incorrect. The question says that the starred stick is on every 50th bar. That means there are more than 50 produced. The problem, then, should be viewed as a choose-and-replace situation.

Krumple wrote:

Why assume? So you are saying "you" are the only one buying? If there are others then it changes the maths and the outcome. So yes its relevant.

Not if the supply is unlimited.

No the question becomes, is the 50th purchased guaranteed a star?

Krumple wrote:

No the question becomes, is the 50th purchased guaranteed a star?

That's a dumb question.

It's simple. There's a 1/50 chance of getting a starred stick any time you purchase an ice cream bar. That means there is a 49/50 (or 98%) chance of not getting a starred stick. The chances of not getting a starred stick when purchasing two bars is .98 * .98 or .98 squared. The chance of not getting a starred stick when purchasing three bars is .98*.98*.98 or .98 cubed, and so on. At .98 to the 35th power, the result dips below .5, meaning that if I buy 35 bars, there's less than a 50% chance that I will not get any starred sticks. Or, to look at it another way, there's a greater than 50% chance that one of them will have a star. Since each bar costs 10 cents, I should expect, on average, to spend approximately $3.50 to get one starred stick.

# - chance of getting no starred sticks
1-0.98
2-0.9604
3-0.941192
4-0.92236816
5-0.903920797
6-0.885842381
7-0.868125533
8-0.850763023
9-0.833747762
10-0.817072807
11-0.800731351
12-0.784716724
13-0.769022389
14-0.753641941
15-0.738569103
16-0.723797721
17-0.709321766
18-0.695135331
19-0.681232624
20-0.667607972
21-0.654255812
22-0.641170696
23-0.628347282
24-0.615780337
25-0.60346473
26-0.591395435
27-0.579567526
28-0.567976176
29-0.556616652
30-0.545484319
31-0.534574633
32-0.52388314
33-0.513405478
34-0.503137368
35-0.493074621

I said the probability for finding the star would go up after every purchase and you said incorrect.

Since you say every 50th has a star the average drops due to the incease in odds of obtaining the star stick after each consecutive purchase.

However if its random then you have 1 in 50 chance for the first purchase with a reduction by one each additional purchase.