5 added 62 characters in body edited Aug 21 '17 at 13:37 ekvall 3,68911 gold badge1111 silver badges3434 bronze badges One reason you're getting different answers from different people is that the question can be interpreted in different ways, and it isn't clear what you mean by "probability" here. One way to make sense of the question is to assign priors and reason using Bayes' rule as in Matthew's answer. Before asking for probabilities, you have to decide what's modeled as random and what's not. It's not universally accepted that unknown but fixed quantities should be assigned priors. Here's a similar experiment to yours that highlights the problem with the question: Assume $$X_i$$, $$i = 1, \dots, 1000$$ are i.i.d. Bernoulli random variables with success probability (mean) $$p = 0.5$$. For interpretability, let's think of the $$X_i$$ as coin flips. Suppose you observe (the sufficient statistic) $$\sum_{i = 1}^{1000}X_i = 900$$. What is the probability that the coin is fair? From a frequentist perspective the question is either nonsensical or the answer is "one". If you're Bayesian maybe you want to assign a prior distribution to $$p$$, in which case the question makes sense. The fundamental difference between my example and the question is that $$p$$ is unknown in the question, and the question disguises the fact that the actual randomness is whether a (presumably randomly sampled) person answers that the car is blue or not. The car's color is not randomly assigned and thus it's uninteresting to speak of the probability of it being blue from a frequentist perspective. One reason you're getting different answers from different people is that the question can be interpreted in different ways, and it isn't clear what you mean by "probability" here. One way to make sense of the question is to assign priors and reason using Bayes' rule as in Matthew's answer. Before asking for probabilities, you have to decide what's modeled as random and what's not. It's not universally accepted that unknown but fixed quantities should be assigned priors. Here's a similar experiment to yours that highlights the problem with the question: Assume $$X_i$$, $$i = 1, \dots, 1000$$ are i.i.d. Bernoulli random variables with success probability (mean) $$p = 0.5$$. Suppose you observe (the sufficient statistic) $$\sum_{i = 1}^{1000}X_i = 900$$. What is the probability that the coin is fair? From a frequentist perspective the question is either nonsensical or the answer is "one". If you're Bayesian maybe you want to assign a prior distribution to $$p$$, in which case the question makes sense. The fundamental difference between my example and the question is that $$p$$ is unknown in the question, and the question disguises the fact that the actual randomness is whether a (presumably randomly sampled) person answers that the car is blue or not. The car's color is not randomly assigned and thus it's uninteresting to speak of the probability of it being blue from a frequentist perspective. One reason you're getting different answers from different people is that the question can be interpreted in different ways, and it isn't clear what you mean by "probability" here. One way to make sense of the question is to assign priors and reason using Bayes' rule as in Matthew's answer. Before asking for probabilities, you have to decide what's modeled as random and what's not. It's not universally accepted that unknown but fixed quantities should be assigned priors. Here's a similar experiment to yours that highlights the problem with the question: Assume $$X_i$$, $$i = 1, \dots, 1000$$ are i.i.d. Bernoulli random variables with success probability (mean) $$p = 0.5$$. For interpretability, let's think of the $$X_i$$ as coin flips. Suppose you observe (the sufficient statistic) $$\sum_{i = 1}^{1000}X_i = 900$$. What is the probability that the coin is fair? From a frequentist perspective the question is either nonsensical or the answer is "one". If you're Bayesian maybe you want to assign a prior distribution to $$p$$, in which case the question makes sense. The fundamental difference between my example and the question is that $$p$$ is unknown in the question, and the question disguises the fact that the actual randomness is whether a (presumably randomly sampled) person answers that the car is blue or not. The car's color is not randomly assigned and thus it's uninteresting to speak of the probability of it being blue from a frequentist perspective. 4 added 576 characters in body edited Aug 21 '17 at 12:43 ekvall 3,68911 gold badge1111 silver badges3434 bronze badges One reason you're getting different answers from different people is that the question can be interpreted in different ways, and it isn't clear what you mean by "probability" here. One way to make sense of the question is to assign priors and reason using Bayes' rule as in Matthew's answer. Another interpretation is the followingBefore asking for probabilities, you have to decide what's modeled as random and what's not. In setting upIt's not universally accepted that unknown but fixed quantities should be assigned priors. Here's a similar experiment to yours that highlights the problem with the question you're ascertaining that: Assume $$X_i$$, $$i = 1, \dots, 1000$$ are i.i.d. Bernoulli random variables with success probability (mean) $$p = 0.5$$. Suppose you observe (the sufficient statistic) $$\sum_{i = 1}^{1000}X_i = 900$$. What is the probability that the coin is fair? From a frequentist perspective the color ofquestion is either nonsensical or the caranswer is well defined and that it is blue"one". Given that contextIf you're Bayesian maybe you want to assign a prior distribution to $$p$$, in which case the probability thatquestion makes sense. The fundamental difference between my example and the carquestion is blue,that $$p$$ is one. The person asked of course doesn't know thisunknown in the question, so can at most sayand the question disguises the fact that the probabilityactual randomness is either one or zero, depending on whether a (presumably randomly sampled) person answers that the car is actually blue or not. The fact that 1000 people were asked about thecar's color is not randomly assigned and couldn't agree doesn't affect the fact thatthus it's uninteresting to speak of the car hasprobability of it being blue from a fixed, but unknown (to the person asked), colorfrequentist perspective. One reason you're getting different answers from different people is that the question can be interpreted in different ways, and it isn't clear what you mean by "probability" here. One way to make sense of the question is to assign priors and reason using Bayes' rule as in Matthew's answer. Another interpretation is the following. In setting up the question you're ascertaining that the color of the car is well defined and that it is blue. Given that context, the probability that the car is blue, is one. The person asked of course doesn't know this, so can at most say that the probability is either one or zero, depending on whether the car is actually blue or not. The fact that 1000 people were asked about the color and couldn't agree doesn't affect the fact that the car has a fixed, but unknown (to the person asked), color. One reason you're getting different answers from different people is that the question can be interpreted in different ways, and it isn't clear what you mean by "probability" here. One way to make sense of the question is to assign priors and reason using Bayes' rule as in Matthew's answer. Before asking for probabilities, you have to decide what's modeled as random and what's not. It's not universally accepted that unknown but fixed quantities should be assigned priors. Here's a similar experiment to yours that highlights the problem with the question: Assume $$X_i$$, $$i = 1, \dots, 1000$$ are i.i.d. Bernoulli random variables with success probability (mean) $$p = 0.5$$. Suppose you observe (the sufficient statistic) $$\sum_{i = 1}^{1000}X_i = 900$$. What is the probability that the coin is fair? From a frequentist perspective the question is either nonsensical or the answer is "one". If you're Bayesian maybe you want to assign a prior distribution to $$p$$, in which case the question makes sense. The fundamental difference between my example and the question is that $$p$$ is unknown in the question, and the question disguises the fact that the actual randomness is whether a (presumably randomly sampled) person answers that the car is blue or not. The car's color is not randomly assigned and thus it's uninteresting to speak of the probability of it being blue from a frequentist perspective. 3 deleted 2 characters in body edited Aug 20 '17 at 21:51 ekvall 3,68911 gold badge1111 silver badges3434 bronze badges One reason you're getting different answers from different people is that the question can be interpreted in different ways, and it isn't clear what you mean by "probability" here. One way to make sense of the question is to assign priors and reason using Bayes' rule as in Matthew's answer. Another interpretation is the following. In setting up the question you're ascertaining that the color of the car is well defined and that it is blue. Given that context, the "probability"probability that the car is blue, is one. The person asked of course doesn't know this, so can at most say that the probability is either zero or one or zero, depending on whether the car is actually blue or not. The fact that 1000 people were asked about the color and couldn't agree doesn't affect the fact that the car has a fixed, but unknown (to the person asked), color. One reason you're getting different answers from different people is that the question can be interpreted in different ways, and it isn't clear what you mean by "probability" here. One way to make sense of the question is to assign priors and reason using Bayes' rule as in Matthew's answer. Another interpretation is the following. In setting up the question you're ascertaining that the color of the car is well defined and that it is blue. Given that context, the "probability" that the car is blue, is one. The person asked of course doesn't know this, so can at most say that the probability is either zero or one, depending on whether the car is actually blue or not. The fact that 1000 people were asked about the color and couldn't agree doesn't affect the fact that the car has a fixed, but unknown (to the person asked), color. One reason you're getting different answers from different people is that the question can be interpreted in different ways, and it isn't clear what you mean by "probability" here. One way to make sense of the question is to assign priors and reason using Bayes' rule as in Matthew's answer. Another interpretation is the following. In setting up the question you're ascertaining that the color of the car is well defined and that it is blue. Given that context, the probability that the car is blue, is one. The person asked of course doesn't know this, so can at most say that the probability is either one or zero, depending on whether the car is actually blue or not. The fact that 1000 people were asked about the color and couldn't agree doesn't affect the fact that the car has a fixed, but unknown (to the person asked), color. 2 added 143 characters in body edited Aug 20 '17 at 21:38 ekvall 3,68911 gold badge1111 silver badges3434 bronze badges 1 answered Aug 20 '17 at 21:28 ekvall 3,68911 gold badge1111 silver badges3434 bronze badges