The model has two states, with uniform initial distribution. The transition probabilities are:

to State 1 | to State 2 | |

From State 1 | 1/3 | 2/3 |

From State 2 | 3/4 | 1/4 |

The output alphabet is {+, 0, -}, with probabilities:

State 1 | State 2 | |

P(+) | 1/4 | 1/3 |

P(0) | 1/2 | 1/3 |

P(-) | 1/4 | 1/3 |

The object is to find the probability of the output sequence + - 0 using the forward algorithm and then again using the backward algorithm.

The forward algorithm starts by multiplying the initial probabilities (1/2) by the probabilities of + (1/4 and 1/3):

State 1 | 1/2 * 1/4 = 1/8 | 1/8 * 1/3 * 1/4 + 1/6 * 3/4 * 1/4 = 1/24 | 1/24 * 1/3 * 1/2 + 1/24 * 3/4 * 1/2 = 13/576 |

State 2 | 1/2 * 1/3 = 1/6 | 1/8 * 2/3 * 1/3 + 1/6 * 1/4 * 1/3 = 1/24 | 1/24 * 2/3 * 1/3 + 1/24 * 1/4 * 1/3 = 11/864 |

The total probability of + - 0 is the sum of these. The common denominator is 1728, so 13/576 + 11/864 = 61/1728

The backward algorithm starts with ones on the right, and moves to the left:

State 1 | 7/18 * 1/3 * 1/4 + 11/24 * 2/3 * 1/3 = 29/216 | 1 * 1/3 * 1/2 + 1 * 2/3 * 1/3 = 7/18 | 1 |

State 2 | 7/18 * 3/4 * 1/4 + 11/24 * 1/4 * 1/3 = 1/9 | 1 * 3/4* 1/2 + 1 * 1/4 * 1/3 = 11/24 | 1 |

In this case, we can't just add these together, we must multiply by the initial probabilities and the probability of producing a + in the appropriate state:

29/216 * 1/2 * 1/4 + 1/9 * 1/2 * 1/3 = 61/1728