A box contains 55 balls numbered from 1 to 55. If 8 balls are drawn with replacement, what is the probability that at least two of them have the same number?
Well, let's put our funny hats on and calculate the probability!
To find the probability, we can calculate the probability of the opposite event and then subtract it from 1.
The opposite event here is that none of the balls have the same number. In this case, on the first draw, we can pick any of the 55 balls. On the second draw, we need to pick any of the remaining 54 balls. And so on, until the eighth draw, where we can pick one out of the remaining 48 balls.
So, the probability of none of the balls having the same number is:
(55/55) * (54/55) * (53/55) * ... * (48/55)
Now, we can subtract this probability from 1 to find the probability we're looking for.
1 - [(55/55) * (54/55) * (53/55) * ... * (48/55)]
And there you go, quote the formula, do the math, and you'll find your answer!
To find the probability that at least two balls have the same number when 8 balls are drawn with replacement, we can use the complement rule.
First, let's find the probability that all 8 balls have different numbers.
For the first ball, there are 55 options.
For the second ball, there are 54 options (one less than the previous ball since we need a different number).
For the third ball, there are 53 options, and so on, until the eighth ball.
So, the probability of all 8 balls having different numbers is given by:
(55/55) * (54/55) * (53/55) * ... * (48/55)
Next, let's find the probability of at least two balls having the same number by using the complement rule.
The complement of the event "at least two balls have the same number" is "all 8 balls have different numbers".
Therefore, the probability of at least two balls having the same number is:
1 - [(55/55) * (54/55) * (53/55) * ... * (48/55)]
Simplifying this expression gives us the probability that at least two balls have the same number when 8 balls are drawn with replacement.
To find the probability that at least two balls have the same number, we can find the probability of the complementary event - that is, the probability that all 8 balls have distinct numbers, and then subtract it from 1.
When drawing with replacement, the probability of drawing a specific number on one draw is 1/55, since there are 55 balls in the box.
To find the probability of drawing 8 distinct numbers, we start with the first draw, which has a probability of 1. Then, for each subsequent draw, we subtract the number of balls from the previous draw that have already been drawn from the total number of balls in the box.
So, for the subsequent draws, the probabilities are multiplied together as follows:
(54/55) * (53/55) * (52/55) * (51/55) * (50/55) * (49/55) * (48/55) * (47/55)
Now, we can calculate the probability of drawing all 8 distinct numbers:
P(distinct) = (54/55) * (53/55) * (52/55) * (51/55) * (50/55) * (49/55) * (48/55) * (47/55) = 0.367
Finally, we subtract the probability of drawing all 8 distinct numbers from 1 to find the probability that at least two balls have the same number:
P(at least two balls with same number) = 1 - P(distinct) = 1 - 0.367 = 0.633
Therefore, the probability that at least two of the 8 drawn balls have the same number is approximately 0.633.
Number of (ordered) ways to draw 8 balls with replacement
= 55^8
Number of ways to draw 8 different balls
=55P8
= 55!/47!
Probability of drawing 8 balls with at least two having the same number
= 1-(55!/47!)/55^8
= 25193936479/60897409375
= 0.414 approx.