Answer of question 2 of HW6:
𝑃(𝑆2 = 𝑟 |𝑆1 = 𝑠, 𝑂2 = 𝑢) =
=
=
=
𝑃(𝑆2 = 𝑟, 𝑆1 = 𝑠, 𝑂2 = 𝑢)
𝑃(𝑆1 = 𝑠, 𝑂2 = 𝑢)
𝑃(𝑆2 = 𝑟, 𝑆1 = 𝑠, 𝑂2 = 𝑢)
∑𝑎∈{𝑟,𝑠} 𝑃(𝑆2 = 𝑎, 𝑆1 = 𝑠, 𝑂2 = 𝑢)
𝑃(𝑂2 = 𝑢 | 𝑆2 = 𝑟, 𝑆1 = 𝑠)𝑃(𝑆2 = 𝑟, 𝑆1 = 𝑠)
∑𝑎∈{𝑟,𝑠,𝑓} 𝑃(𝑂2 = 𝑢 | 𝑆2 = 𝑎, 𝑆1 = 𝑠)𝑃(𝑆2 = 𝑎, 𝑆1 = 𝑠)
𝑃(𝑂2 = 𝑢 | 𝑆2 = 𝑟, 𝑆1 = 𝑠)𝑃(𝑆2 = 𝑟|𝑆1 = 𝑠)𝑃(𝑆1 = 𝑠)
∑𝑎∈{𝑟,𝑠,𝑓} 𝑃(𝑂2 = 𝑢 | 𝑆2 = 𝑎, 𝑆1 = 𝑠)𝑃(𝑆2 = 𝑎|𝑆1 = 𝑠)𝑃(𝑆1 = 𝑠)
=
𝑃(𝑂2 = 𝑢 | 𝑆2 = 𝑟)𝑃(𝑆2 = 𝑟 | 𝑆1 = 𝑠)𝑃(𝑆1 = 𝑠)
∑𝑎∈{𝑟,𝑠,𝑓} 𝑃(𝑂2 = 𝑢 | 𝑆2 = 𝑎)𝑃(𝑆2 = 𝑎 | 𝑆1 = 𝑠)𝑃(𝑆1 = 𝑠)
=
=
𝑃(𝑂2 = 𝑢 | 𝑆2 = 𝑟)𝑃(𝑆2 = 𝑟 | 𝑆1 = 𝑠)
∑𝑎∈{𝑟,𝑠,𝑓} 𝑃(𝑂2 = 𝑢 | 𝑆2 = 𝑎)𝑃(𝑆2 = 𝑎 | 𝑆1 = 𝑠)
0.8 × 0.05
8
=
≅ 0.2424
0.8 × 0.05 + 0.1 × 0.8 + 0.3 × 0.15 33
Note: Mentioning the 𝑃(𝑂1 = 𝑢) is irrelevant.