Question

Many people believe that they can tell the difference between Coke and Pepsi. Other people say that the two brands​ cannot be distinguished. To test​ this, a random sample of 20 adults was selected to participate in a test. After being​ blindfolded, each person was given a small taste of either Coke or Pepsi and asked to indicate which brand soft drink it was. If people really​ can't tell the​ difference, what is the probability that fewer than 6 people will guess​ correctly

Answer 1

Answer:0.0206

Explanation:

Using Binomial distribution for a sample of 20 adults

Let r denotes the no of correct answers out of 20

Probability that fewer than 6 people will guess correctly is P(r<6)

P(r<6)=P(r=0)+P(r=1)+P(r=2)+P(r=3)+P(r=4)+P(r=5)

`P(r=0)=^(20)C_0\left ( 0.5\right )^(0)\left ( 0.5\right )^(20)=\left ( 0.5\right )^(20)`

`P(r=1)=^(20)C_0\left ( 0.5\right )^(1)\left ( 0.5\right )^(19)=20\left ( 0.5\right )^(20)`

`P(r=2)=^(20)C_0\left ( 0.5\right )^(2)\left ( 0.5\right )^(18)=190\left ( 0.5\right )^(20)`

`P(r=3)=^(20)C_0\left ( 0.5\right )^(3)\left ( 0.5\right )^(17)=1140\left ( 0.5\right )^(20)`

`P(r=4)=^(20)C_0\left ( 0.5\right )^(4)\left ( 0.5\right )^(16)=4845\left ( 0.5\right )^(20)`

`P(r=5)=^(20)C_0\left ( 0.5\right )^(5)\left ( 0.5\right )^(15)=15,504\left ( 0.5\right )^(20)`

`P(r<6)=\left ( 0.5\right )^(20)\left [ 1+20+190+1140+4845+15504\right ]`

`P(r<6)=\left ( 0.5\right )^(20)* 21,700`

P(r<6)=0.02069