A fair die is rolled 12 times. Consider the following four possible outcomes:(i) 5 2 6 3 2 1 4 1 6 5 3 4(ii) 1 1 2 2 3 3 4 4 5 5 6 6(iii) 6 6 6 6 6 6 6 6 6 6 6 6(iv) 1 5 4 3 5 1 2 4 4 6 4 5Which of the following is the most likely outcome: (i), (ii), (iii), (iv)? (i) because number of die outcomes (1, 2, 3, 4, 5, 6) is equal but in a random order. (ii) because the number of die outcomes (1, 2, 3, 4, 5, 6) is equal. (iii) because the number 6 is just as likely as any other number on a die. (iv) because you won’t necessarily get the same number of of die outcomes (1, 2, 3, 4, 5, 6) with a fair die. They are all equally likely.
Question
A fair die is rolled 12 times. Consider the following four possible outcomes:(i) 5 2 6 3 2 1 4 1 6 5 3 4(ii) 1 1 2 2 3 3 4 4 5 5 6 6(iii) 6 6 6 6 6 6 6 6 6 6 6 6(iv) 1 5 4 3 5 1 2 4 4 6 4 5Which of the following is the most likely outcome: (i), (ii), (iii), (iv)? (i) because number of die outcomes (1, 2, 3, 4, 5, 6) is equal but in a random order. (ii) because the number of die outcomes (1, 2, 3, 4, 5, 6) is equal. (iii) because the number 6 is just as likely as any other number on a die. (iv) because you won’t necessarily get the same number of of die outcomes (1, 2, 3, 4, 5, 6) with a fair die. They are all equally likely.
Solution
The question is asking which of the four outcomes is the most likely when a fair die is rolled 12 times.
A fair die has six faces, each with an equal probability of being rolled. Therefore, each individual roll is independent and has a 1/6 chance of landing on any given number (1, 2, 3, 4, 5, or 6).
(i) This outcome has each number from 1 to 6 appearing twice, but in a random order. The probability of this specific sequence occurring is (1/6)^12, because each roll is independent.
(ii) This outcome also has each number from 1 to 6 appearing twice, but in a specific order. The probability of this specific sequence occurring is also (1/6)^12, for the same reason as above.
(iii) This outcome has the number 6 appearing 12 times in a row. The probability of this specific sequence occurring is also (1/6)^12, because each roll is independent.
(iv) This outcome has the numbers 1 to 6 appearing a different number of times, in a random order. The probability of this specific sequence occurring is also (1/6)^12, for the same reason as above.
Therefore, all four outcomes are equally likely, because the probability of each specific sequence occurring is the same.
Similar Questions
A fair coin is tossed 12 times. Which of the following outcomes (i, ii, iii, or iv) is most likely?(i) H T H T H T H T H T H T(ii) H T T H H T T H T H H T(iii) H H H H H H H H H H H H(iv) T T T H T H H H H T H H (i) because there are an equal number of heads and tails. (ii) because there are an equal number of heads and tails but in a random order (iii) because heads are just as likely as tails (iv) because you won’t necessarily get the same number of heads and tails with a fair coin They are all equally likely.
Below is a list of all possible outcomes in the experiment of rolling two die.Note: The table gives the values of each die, not the sum of the two die.Determine the following probabilities. Write your answers as reduced fractions.P𝑃(sum is 12) = P𝑃(sum is 8) = P𝑃(sum is 4) =
Three identical dice are rolled. The probability that the same number willappear on each of them is:(a) 1/6(b) 1/12(c) 1/36(d) 1
Two dice are rolled and the sum is recorded.(a) List all possible outcomes (i.e. the sample space).(b) Calculate the probability of rolling:(i) A sum of 5(ii) A sum of 4 or less(iii) A sum of at most 11(iv) A sum that is a multiple of 3 or a multiple of 4
a) If you roll a single die and count the number of dots on top, what is the sample space of all possible outcomes? Are the outcomes equally likely?1, 2, 3, 4, 5, 6; not equally likely0, 1, 2, 3, 4, 5, 6; not equally likely 1, 2, 3, 4, 5, 6; equally likely0, 1, 2, 3, 4, 5, 6; equally likely(b) Assign probabilities to the outcomes of the sample space of part (a). (Enter your answers as fractions.)Outcome Probability1 2 3 4 5 6 Do the probabilities add up to 1? Should they add up to 1? Explain.Yes, but they should not because these values do not cover the entire sample space.Yes, because these values cover the entire sample space. No, because these values do not cover the entire sample space.No, but they should because these values cover the entire sample space.(c) What is the probability of getting a number less than 2 on a single throw? (Enter your answer as a fraction.)(d) What is the probability of getting 1 or 2 on a single throw? (Enter your answer as a fraction.)
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.