A spinner can land in any of four positions, A, B, C, and D, with equal probability. The spinner is used twice, and the position is noted each time. Let the random variable Y denote the number of positions on which the spinner did not land. Compute the probabilities for each value of Y .

Step 1 of 3:

Given that a spinner can land in any four positions, A, B, C, D, with equal probability.

The spinner is used twice, then the sample space is given by

S = { AA, AB, AC, AD, BA, BB, BC, BD, CA, CB, CC, CD, DA, DB, DC, DD}

Let Y denote the number of positions on which the spinner did not land.

Step 2 of 3:

Here we need to compute the probabilities for each value of Y.

Now,

Sample point |
Y |
Sample point |
Y |

AA |
3 |
CA |
2 |

AB |
2 |
CB |
2 |

AC |
2 |
CC |
3 |

AD |
2 |
CD |
2 |

BA |
2 |
DA |
2 |

BB |
3 |
DB |
2 |

BC |
2 |
DC |
2 |

BD |
2 |
DD |
3 |

Each position has probability so every ordering of two positions (from two spins) has probability The values for Y are 2, 3.

Then,

{Y = 2} = {AB, AC, AD, BA, BC, BD, CA, CB, CD, DA, DB, DC}

{ Y = 3} = { AA, BB, CC, DD}.

Hence,

P{Y = 2} =