Question

Which experiment describes a binomial experiment?

Draw 4 cards from a deck of cards, without replacement, and record each time the card is an ace.

Spin a spinner with sections labeled A through F 6 times and record each time you get a vowel.

Roll 2 dice until you get a sum of 8 and record how many trials you made.

Ask 9 people the same question and record each time they answered yes, no, or maybe.

Answer 1

This is the definition of a binomial experiment:
a probability experiment in which there are a fixed number of independent trials, each outcome for each trial is labeled either success or failure (where success and failure are complementary events) and the probability p of success is the same for each trial; the probability of failure for each trial is 1 minus p

Answer 2

I have not learned about binomial experiments yet but you can try asking @Destinymasha or @petiteme

Answer 3

I think the answer is B because there is a specific number of trials, one spin does not affect the other, and the spins can be labeled as successes of failures.

Answer 4

You can recognize a binomial experiment in that:
1) There can be only one of TWO possible outcomes.
2) Every outcome has the same probability as every other one.
3) Every outcome is independent of every other one.

So: which of the answer choices is the correct one?

Answer 5

Welp..A binomial experiment is where you perform a certain number of trials, and count the number of successes

Answer 6

I think it is B.

Answer 7

Please defend your answer choice. What was your rationale? Did you look carefully at the 3 characteristics of a binomial experiment that I typed out for you?

Answer 8

I'm not sure how it could be any other answer. It can't be A because the cards are not being replaced. It can't be C because there are not a fixed number of trials. I don't think it can be D because there are 3 options to choose from; it's not just successes and failures.

Answer 9

I think it is B too.
Only two possible outcomes: either it is a vowel or it is not
Probability of getting a vowel out of A,B,C,D,E and F is: 2/6 = 1/3 and the probability remains the same for each spin.
Each spin is independent of any other spin.

In A, the probability of getting an ace keeps on changing because the withdrawn cards are NOT replaced.

Similar reasoning to rule out the rest. The last one has 3 possible outcomes and hence it is not binomial.

Answer 10

That is what I was thinking. Great explanation! Thanks.

Answer 11

You are welcome.

Answer 12

Could I ask you another question? @ranga

Answer 13

BTW, I would add one more condition to the list given by mathmale to determine if an experiment is binomial. In a binomial experiment, the number of trials should be fixed. In C) the number of trials is NOT fixed and so it is not a BE.

Go ahead with your next question. I will try.

Answer 14

Yes I knew the trials had to be fixed and that is how I ruled out C.

Here is the question:
Which situation shows the appropriate use of the law of large numbers?
Wilma predicts rolling a 2 on her 120th roll because she rolled a 2 a total of 19 times already.
Cindy predicts rolling a 2 about 30 times if she rolls a die 100 times.
Amos predicts rolling a 2 about 40 times if he rolls a die 240 times.
Keith predicts rolling a 2 one time if he rolls a die 6 times.

Answer 15

Here is a definition for law of large numbers:
The law of large numbers states that as the number of independent trials increases, it becomes more likely that the experimental probability of an event gets closer to its theoretical probability.

Answer 16

When there is LARGE number of experimental trials, the observed probability will get closer to the theoretical probability.

Answer 17

"Wilma predicts rolling a 2 on her 120th roll"
The 120th roll is independent of any previous or future rolls. The probability of getting a 2 in the 120th roll is same as the probability of getting a 2 in any other roll and is independent oh how many 2's she got before.

You try the next choice.

Answer 18

Cindy predicts rolling a 2 about 30 times if she rolls a die 100 times.
100 is a large number of trials and the trials are independent. The theoretical probability of rolling a 2 in 100 rolls in 1/2. Would B be the correct answer?

Answer 19

The probability of rolling a 2 in a fair, single die is 1/6.

Answer 20

So if Cindy rolls a die 100 times she can expect to get 2 about 100 * 1/6 = 16 or 17 times and NOT 30 times.

Answer 21

Or, to get to the answer quickly, the probability of getting a 2 when rolling a fair, single die is 1/6 (theoretical probability)
When the number of trials is large such as in C), you can expect to get 2 about 40 times in 240 trials because 240 * 1/6 = 40.

Answer 22

Oh that makes sense! So it is C because if you roll the dice 240 (240 * 1/6) you get 40.

Answer 23

Correct.

In the last case, he rolls the die just 6 times. Even though 6 * 1/6 = 1 you cannot really expect the experimental probability to match the theoretical probability because the number of trails is so low. A die has 6 possibilities and he rolls it only 6 times. It is too low a number to apply theoretical probability to an experimental observation. But C) has 240 trials. That is a sufficiently large number that the number of 2's can be close to 40.

Answer 24

Thanks for the awesome explanations! I really appreciate your help. :)

Answer 25

I also understand it a lot better.

Answer 26

You are very welcome.