What is the answer?

Question

In a bingo game, balls numbered 1-25 are put in a bin. Balls are randomly drawn and not replaced. What is the probability of selecting 5 two digit numbers and selecting 5 odd numbers?

Answer 1

The question seems to ask for both probabilities, separately. If that is the question, then we shoud first, let Event A be (all 5 balls being 2 digit numbers) and Event B = (all 5 balls being odd).

The probability would have to allow for the condition that the ball chosen each time will not be replaced.

So, for P(A), in the sample space of all 25 numbers, there are 9 balls that are single digit and 16 balls that are double digit.

P(A1) = P(1st Ball picked and is double digit) = 16/25

P(A2) = P(2nd Ball is picked and is double digit, given that the 1st Ball has been taken out) = 15/24

therefore, with that reasoning,

P(A3) = 14/23

P(A4) = 13/22

P(A5) = 12/21

Multiplying these event probabilities would be 524160/6375600 or 312/3795 or 0.0822 or 8.22%

The Probability of B, in the same way, would have 13 of the possible outcomes that are odd and 12 that are even.

So, P(B1) = P(1st Ball is odd) = 13/25

P(B2) = P(2nd Ball is odd, given that the 1st Ball has been removed) = 12/24

P(B3) = 11/23

P(B4) = 10/22

P(B5) = 9/21.

Multiplying these event probabilities would be 154440/6375600 or 39/1610 or 0.0242 or 2.42%

If the question meant that the 2 events had to be true at the same time, both double digit and odd, then the probabilities would be much lower, as there are only 8 odd double digit numbers of the 25 numbers.

The P(A and B) = P(both A and B events happening) would be found as below:

P(AB1) = 8/25

P(AB2) = 7/24

P(AB3) = 6/23

P(AB4) = 5/22

P(AB5) = 4/21

P(A and B) = 6720/6375600 or 4/3795 or 0.0011 or 0.11%

Answer 2

Hi Emani,

The solution you are looking for is 0.1584. The event that we want is P(5 2-digit AND 5 odd). Using the formula: P(A AND B) = P(A) + P(B) - P(A)P(B).

The probability of event A is: (16x15x14x13x12)/(25x24x23x22x21) = 0.0822. This is because there are 16 2-digit balls, and we have to count down because the balls are taken without replacement.

The probability of event B is: (13x12x11x10x9)/(20x19x18x17x16) = 0.0830. There are 13 odd numbered balls, and we still need to count down, because the balls are not replaced.

Plugging them into the formula yields: 0.0822 + 0.0830 - (0.0822)(0.0830) = 0.1584.

I hope this helps you,

-Kevin