Iggy's Probability Quiz 2

Quiz Answer Key and Fun Facts

1. You have 4 coins. Three of the coins are normal, but one of them is heads on both sides. You pick a coin at random without looking. The coin you pick has heads on one side. What are the odds that if you flip the coin over, the other side will be tails?

Answer: 3/5

There are 8 sides on the 4 coins. Heads is on 5 sides while tails is on 3 sides. Out of the 5 sides that are heads, 3 of them have tails on the other side. This means the odds are 3 out of 5.

2. If you roll three normal 6-sided dice, what are the odds that the dice rolls add up to 16?

Answer: 1/36

If you do 6 to the 3rd power, there are 216 possible results from rolling three dice. The possible results that total 16 are 4-6-6, 6-6-4, 6-4-6, 5-5-6, 5-6-5, and 6-5-5. Since there are 6 possible results, the odds are 6/216 or 1/36.

3. If you randomly flip 5 normal coins, then what is the probability that you roll 3 tails and 2 heads?

Answer: 5/16

If you do 2 to the 5th power, then there are 32 possible results from flipping 5 coins. The possible results containing 3 tails and 2 heads, are: TTTHH, TTHTH, TTHHT, THHTT, THTHT, THTTH, HHTTT, HTHTT, HTTHT, and HTTTH. Since there are 10 possible results, the probability is 10/32 or 5/16.

4. If you randomly guessed on the last three multiple choice problems, then what is the probability you got at least one of them correct?

Answer: 37/64

First I'll figure out the probability you didn't get any correct. There are 4 choices and 3 wrong ones in each question, so the odds you will get a question wrong are 3/4. The odds you will get all three wrong are 3/4 * 3/4 * 3/4 = 27/64. This means that 37/64 are the odds you will not get them all wrong. Not getting them all wrong and getting at least one correct are the same thing.

5. The answer to a fill-in-the-blank question is a five-digit number that is a palindrome (a number that reads the same backwards and forwards). If you make an educated guess based on the fact the 5-digit number is a palindrome, then the odds of getting the question right are 1 out of what number? The answer is just a numeral like 52 or 420 or 62848112.

Answer: 900

The first number has 9 possibilities from 1-9, so the odds of getting the first digit correct are 1/9. The second digit has 10 possibilities from 0-9, so the odds of getting it correct are 1/10. The odds of getting the third digit correct are also 1/10.

The last two digits are the same as the first two digits so we don't need to consider them. If you multiply 1/9 * 1/10 * 1/10 then there is a 1/900 chance of getting the question correct.

6. If you randomly guessed on 5 questions with 4 answer choices each, then what is the probability you got exactly two correct?

Answer: 135/512

The probability of getting two specific problems correct are 1/4 * 1/4 * 3/4 * 3/4 * 3/4 = 27/1024
There are 32 possible results from answering 5 questions, and the possible results from answering 2/5 questions correct are: WWWCC, WWCWC, WWCCW, WCCWW, WCWCW, WCWWC, CCWWW, CWCWW, CWWCW, and CWWWC.
Each of those 10 possibilities has the same 27/1024 chance of occurring. This means the odds of one of them occurring are 10 times the odds of a specific one occurring. 27/1024 * 10 = 270/1024 or 135/512.

7. A game show requires you to randomly pick 1 of 6 envelopes. The host has hidden a $100 bill in one envelope, and a $1 bill in each of the other 5. Once you've picked, the host is required to remove 2 $1 envelopes you didn't pick. You now have a choice of keeping your original envelope, or paying $2 to switch your choice to one of the 3 you didn't pick. If you choose to pay $2 and switch your choice, then what are your odds of losing money?

Answer: 13/18

There is a 1/6 chance that you originally picked the correct envelope. After the host removes 2 $1 envelopes, there are only 4 envelopes left, but your odds of having the $100 envelope are still 1 in 6. There is a 5/6 chance that the $100 is in one of the 3 remaining envelopes you didn't pick.

This means there is a (5/6)/3 or 5/18 chance that you will win by switching your choice. The odds of losing money by switching your choice are 13/18 since the odds of winning are 5/18. You are likely to lose money if you switch your choice, but you are also 1.66666667 times more likely to get the $100 bill

8. Let's say conceiving a girl is twice as likely to happen as conceiving a boy. If you have 3 children based on those odds, what would be the probability that you have two boys and a girl?

Answer: 2/9

The odds of conceiving a girl and boy are 2/3 and 1/3. This means that the odds of having two boys and a girl in a specific order would be 2/3 * 1/3 * 1/3, which is 2/27. There are three possible combinations of 2 boys and a girl, each with the same probability of 2/27.

This means the odds of one result occurring are three times the odds of a specific result occurring. 2/27 times 3 is 6/27 or 2/9, which is correct.

9. A toy factory generates cubes with a random color (red or blue) on each side. You buy 7 of the cubes and all your cubes have a different number of blue squares. You select one of your cubes at random and happen to pull up a cube showing only one side, which is red. What is the probability that at least two other squares are also red?

Answer: 6/7

Since no two cubes have the same number of each color, there are 21 (0+1+2+3+4+5+6) blue squares in all. Consequently there are also 21 red squares. The odds of choosing the completely red cube are 6 in 21 since there are 6 different red squares you could've picked.

The probability of choosing the cube with 5 red sides is only 5 in 21 since now there are only 5 red squares you could've picked. The odds of choosing a cube with X red sides will be X in 21 where X is a number from 0 to 6. For there to be at least two other red sides, you had to have picked a cube with at least 3 red sides.

The odds will be (3+4+5+6)/21, which is 18/21 or 6/7.

10. You randomly answer 4 multiple choice problems with 4 choices each. What is the probability that you get at least two correct?

Answer: 67/256

There is a (3/4)(3/4)(3/4)(3/4) or 81/256 chance you won't get any questions right. There is a (3/4)(3/4)(1/4)(3/4) or 27/256 chance you will get one specific question right. There are 4 possible combinations of ways you could get one question right, so the total probability is 4(27/256) or 108/256.

The odds you will get zero or one correct are 108/256 + 81/256 = 189/256. The other 67/256 is the probability you will get at least 2 correct.

Source: Author iggy4

This quiz was reviewed by FunTrivia editor crisw before going online.
Any errors found in FunTrivia content are routinely corrected through our feedback system.

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