Theoretical Probability

Question

Warren has 5 standard number cubes. He plans to roll all of them at once. Determine the theoretical probability that exactly 3 number cubes land on an even number. Express as a fraction in simplest form

Answer 1

Warren has 5 standard number cubes. He plans to roll all of them at once. Determine the theoretical probability that exactly 3 number cubes land on an even number. Express as a fraction in simplest form.

What are you solving for:

The probability of rolling exactly three even numbers when rolling five standard number cubes.

Information outline in the problem:

1. Number of dice: 5
2. All dice has 6 sides (1-6)
3. Number of rolls needed to be even: 3

The probability of an event is the number of favorable out comes (divided by) the total number of possible outcomes.

The probability of rolling an even number on a standard die is 1/2.

The probability of rolling an odd number on a standard die is 1/2.

Binomial probability formula is P(x) = (n) p^x (1-p)^n-x

1. n is the number of trials
2. x is the number f successes
3. p is the probability of success on a sngle trial

Step 1:

Calculate the probability of getting exactly 3 even numbers:

1. # of trials n =5
2. # of successes x = 3
3. The probability of success ( roll an even number) p = 1/2
4. The probability of failure ( roll an odd number) p = 1 - 1/2

Use the Binomial probability formula:

P(3) = (5) (1/2)^3 (1/2)^5-3

(3)

P(3) = (5) (1/2)^3 (1/2)^2

(3)

Step 2:

Calculate the binomial coefficient.

(5) = 5! = 5! = 5 x 4 = 20/2 = 10

(3) 3!(5-3)! 3!2! 2 x 1

Step 3:

P(3) = 10 (1/2)^3 (1/2)^2

10 (1/2) ^3 = 1/8 (1/2)^2 = 1/4

P(3) = 10 (1/8) (1/4) 10 x (1/8 x 1/4) = 10/1 x 1/32 = 10/32

P(3) = 10 (1/32)

P(3) = 10/32

Step 4:

Simplify the fraction.

P (3) = 10/32 = 5/16 10 divided by 2 = 5/16

32 divided by 2

Final Answer:

The probability of rolling exactly 3 even numbers is 5/16 (Express as a fraction in simplest form.)

I hope the mathematical calculations (step by step approach) was helpful. Please let me know if you require any more assistance. If anyone in my neighborhood is interested in setting up an in-person math tutoring session. I look forward to hearing from them. Have an amazing day. Doris H.

Answer 2

I have provided the solution in this video in 4 easy steps. I hope you find this to be helpful. If you have any additional questions, please feel free to reach out! Take care!

Dr. Christal-Joy Turner

Answer 3

Rolling die, but only asking whether the number is even or odd, is mathematically equivalent to just tossing a coin.

The probability that exactly 3 will be even is equal to the number of ways that could happen, multiplied by the probability of it happening.

The number of ways: 5 choose 3 = 10

(Verification: EEEOO, EEOEO, EEOOE, EOEEO, EOEOE, EOOEE, OEEEO, OEEOE, OEOEE, OOEEE. That's 10 different ways, assuming E denotes even and O denotes odd)

The probability: (1/2)^5 = 1/32

So the probability is 10/32 = 5/16 = 0.3125 = 31.25%