in a box there are 10 red marbles, 20 blue marbles and 30 yellow marbles. If you pick 10 marbles at random without looking, what are the odds that at least half of them are blue?
user
Sure, here's how you can approach solving this problem:

1. First, we need to determine how many ways we can select 10 marbles from a total of 60 (10 red, 20 blue, and 30 yellow) marbles. This is a combination problem, so we'll use the formula:

   Combinations = (Number of items)! / [(Number of items to be selected)! * (Number of items left untouched)!]

   In this case, (Number of items)! = 60!, (Number of items to be selected)! = 10!, and (Number of items left untouched)! = (60 - 10)!, which simplifies to 50!.

   Thus, Combinations = 60! / [10! * (50 - 10)!], which simplifies to 302,580,640, which is a large number.

2. Next, we need to count the number of times we get at least half (5) blue marbles in our selection of 10. The simplest way to do this is to count the total number of combinations and then subtract the number of combinations where we have less than 5 blue marbles.

   Let's break it down:

   - Combinations with less than 5 blue marbles: For this, we need to count the number of combinations where we have 0, 1, 2, 3 or 4 blue marbles. This can be done using a nested binomial coefficient, where "blue" represents the event of selecting a blue marble, and "not blue" represents the event of not selecting a blue marble.

     - Combinations where we pick 0 blue marbles: 10C0 * 20C0 * 30C10 = 1 * 1 * 5090 = 5090.

     - Combinations where we pick 1 blue marble: 10C1 * 20C9 * 30C9 = 10 * 191 * 243 = 45,180.

     - Combinations where we pick 2 blue marbles: 10C2 * 20C8 * 30C8 = 45 * 209 * 3628 = 2,215,600.

     - Combinations where we pick 3 blue marbles: 10C3 * 20C7 * 30C7 = 120 * 14 * 210 = 33,616,000.

     - Combinations where we pick 4 blue marbles: 10C4 * 20C6 * 30C6 = 70 * 10 * 498 = 144,640,000.

   - Total combinations (including at least 5 blue marbles): We can count this by using the binomial coefficient C(n, k) in a larger formula, which is a generalization of C(n, k). In this case, n = 20, k = 5, r = 10. The total number of ways to pick r objects from a set of n objects when at least k are selected is C(n-1, k-1) * C(n, r), where n >= k.

     So, the total number of ways is C(19, 4) * C(20, 5) = 13,610,000 * 1,043,490,912 = 14,050,986,160,000.

3. The probabilities (odds) of getting at least 5 blue marbles is the ratio of this number and the total number of combinations (step 1), which gives approximately 2.278 billion.

So, your odds of getting or more blue marbles (5 or 6 or 7 or 8 or 9 or 10) are approximately 2.278 billion/302,580,640, or about 7.2 times.

Therefore, you'll have more than half the chances.

I hope this helps to solve your problem. Let me know if you have any other questions.
assistant
in a box there are 10 red marbles, 20 blue marbles and 30 yellow marbles. If you pick 10 marbles at random without looking, what are the odds that at least half of them are blue?
user
To solve this problem, we need to calculate the probability of picking at least 5 blue marbles out of 10. This involves calculating the probabilities for 5, 6, 7, 8, 9, and 10 blue marbles and then adding them together.

The total number of marbles is 60 (10 red + 20 blue + 30 yellow), and the total number of ways to pick 10 marbles out of 60 is given by the combination formula C(n, k) = n! / [k!(n-k)!], where n is the total number of items, k is the number of items to choose, and "!" denotes factorial.

So, the total number of ways to pick 10 marbles is C(60, 10).

The number of ways to pick k blue marbles and 10-k marbles that are not blue is given by the product of two combinations: C(20, k) * C(40, 10-k), where 20 is the total number of blue marbles, 40 is the total number of non-blue marbles (10 red + 30 yellow), k is the number of blue marbles, and 10-k is the number of non-blue marbles.

The probability P(k) of picking k blue marbles and 10-k non-blue marbles is then:

P(k) = [C(20, k) * C(40, 10-k)] / C(60, 10)

The probability of picking at least 5 blue marbles is then:

P(at least 5) = P(5) + P(6) + P(7) + P(8) + P(9) + P(10)

This calculation involves large factorials and is best done with a computer or calculator that can handle large numbers. The final result will be a probability between 0 and 1, which can be converted to a percentage by multiplying by 100.
assistant