In probability can the top number be higher than the bottom?

Question

say the chances of winning the lottery is 1/50, and i play it each week for a whole year. that would mean my chances would be 52/50 . that's greater than 'one'. but there is still a chance i could never win so how is this possible?, its like saying theres a 104% chance i'll win.

Answer 1

Not quite.. Try calculating the chances for NOT winning 52 times in a row.

This is 49/50 or 98% or 0,98 ^ 52. This equals 0,34974856 or 34,97%

So your chance of winning at least once is 100% - 34,97% = 65,03% in that case.

Answer 2

CheGuevara's approach is proper... you have about a 65% chance of winning at least once over 52 weeks

The short answer is no, you can never have a probability > 1

probability of an event is the # ways that event can happen / total number of all events

and you can never have more ways for one event to happen then ways that any event can happen

Edit:  good point IronDuke...  the ODDS of winning at least once is indeed greater than 1

(approximately 65 / 35, or 13 / 7, or almost 2/1)... odds are "successes" / "failures", or "event" / "not event"

Answer 3

Probability is always the number of *good* outcomes divided by the *total* possible outcomes.  Good outcomes are always a subset of total outcomes, so probability can never be more than 1.

Addressing your example with the lottery, the probability over a time period is not the *sum* of the individual probabilities each week.

For example, the chance of flipping heads after two tosses is *not* 1/2 + 1/2.  Instead, it is 1 minus the chance of tossing two tails in a row.

= 1 - (1/2 * 1/2)

= 1 - 1/4

= 3/4

And the chance of flipping at least one head after three tosses is *not* 1/2 + 1/2 + 1/2.  Instead it is:

1 - (1/2)^3

= 1 - 1/8

= 7/8

Similarly with the lottery, if the probability of winning one week is 1/50, then the chance of losing is 49/50.

Over 52 weeks (a year) the odds of losing every week would be:

(49/50)^52

= 0.349748561 (approx. 35%)

Thus the odds of winning the lottery (at least once) after a year of buying tickets would be:

1 - (49/50)^52

= 0.650251439 (approx. 65%)

Answer 4

No, it's not possible. The numerator can never be greater than the denominator, or else the answer will be greater than 1, which defies the basic concept of probability 0 <= P(A) <= 1, where P(A) is the probability of an event A.

I hope that helps. :)

me07.

P.S. Your chances remain the same always. If your chances are 1/50 today, they'll be 1/50 for the next week, for the next year and for infinity.  

Answer 5

The short answer is no, the top number must always be equal to or less than the bottom number.  (All probability is between 0 and 100%, or between 0 and 1 if you're talking about a fraction.)

You chance of winning the lottery each week is independent.  (The lottery numbers drawn one week have no relationship to the numbers drawn the next week.)

So, your chances of winning the lottery are 1/50 for each week.  You can't just add the weeks together.

Since each week is a "brand new contest" (so to speak), the chances stay the same.

Answer 6

Probability cannot be > 1. Odds can be. A horse can be a 2/1 favorite. The odds of winning get worse as the numerator increases.