How to calculate the probability

[Masaru]

I would appreciate if someone can help me with the question below:


When Boris goes to school in winter, the probability that he wears a hat is 5/8. If he wears a hat, the probability that he wears a scarf is2/3. If he does not wear a hat, the probability that he wears a scarf is 1/6. If Boris wears a hat and a scarf, the probability that he wearsgloves is 7/10. Calculate the probability that Boris does not wear all three of hat, scarf and gloves.
I find it very confusing because the information given in the question seems to be totally insufficient to find the solution.
The probability of wearing a hat, scarf and gloves is 5/8 X 2/3 X 7/10 = 7/24, and the probability of wearing a hat, scarf, but no gloves is 5/8 X 2/3 X 3/10 = 1/8, however, I cannot find any other pobabilities namely "hat, no scarf, gloves", "hat, no scarf, no gloves", "no hat, scarf, gloves", "no hat, scarf, no gloves", "no hat, no scarf, gloves" so consequently, I cannot find the probability of "no hat, no scarf, no gloves"

 

[pka]

When Boris goes to school in winter, the probability that he wears a hat is 5/8. If he wears a hat, the probability that he wears a scarf is2/3. If he does not wear a hat, the probability that he wears a scarf is 1/6. If Boris wears a hat and a scarf, the probability that he wearsgloves is 7/10. Calculate the probability that Boris does not wear all three of hat, scarf and gloves.

You simply want to find \(\displaystyle \mathcal{P}(\overline{H\cap G\cap S})\) i.e. the complement of all three.

 

[HallsofIvy]

I would appreciate if someone can help me with the question below:


When Boris goes to school in winter, the probability that he wears a hat is 5/8. If he wears a hat, the probability that he wears a scarf is2/3. If he does not wear a hat, the probability that he wears a scarf is 1/6. If Boris wears a hat and a scarf, the probability that he wearsgloves is 7/10. Calculate the probability that Boris does not wear all three of hat, scarf and gloves.
I find it very confusing because the information given in the question seems to be totally insufficient to find the solution.
The probability of wearing a hat, scarf and gloves is 5/8 X 2/3 X 7/10 = 7/24, and the probability of wearing a hat, scarf, but no gloves is 5/8 X 2/3 X 3/10 = 1/8, however, I cannot find any other pobabilities namely "hat, no scarf, gloves", "hat, no scarf, no gloves", "no hat, scarf, gloves", "no hat, scarf, no gloves", "no hat, no scarf, gloves" so consequently, I cannot find the probability of "no hat, no scarf, no gloves"

The probability that he does NOT wear all three is 1 minus the probability that he does wear all three: 1- 7/24.

 

[soroban]

Hello, Masaru!

Conditional probability: .\(\displaystyle P(A\,|\,B) \:=\:\dfrac{P(A \wedge B)}{P(B)}\)

When Boris goes to school in winter, the probability that he wears a hat is 5/8. [1]
If he wears a hat, the probability that he wears a scarf is 2/3. [2]
If he does not wear a hat, the probability that he wears a scarf is 1/6. [3]
If Boris wears a hat and a scarf, the probability that he wears gloves is 7/10. [4]
Calculate the probability that Boris does not wear all three of hat, scarf and gloves.

From [1], we have: .\(\displaystyle P(H) \,=\,\frac{5}{8},\quad P(\sim\!H) \,=\,\frac{3}{8}\)


From [2], we have: .\(\displaystyle P(S\,|\,H) \,=\,\dfrac{P(S\wedge H)}{P(H)}\,=\,\frac{2}{3} \)

. . \(\displaystyle P(S\wedge H) \:=\:\frac{2}{3}P(H) \:=\:\frac{2}{3}\cdot\frac{5}{8} \quad\Rightarrow\quad P(S\wedge H) \:=\:\frac{5}{12}\)


From [4], we have: .\(\displaystyle P(G\,|\,S\,\wedge H) \:=\:\dfrac{P(G\wedge S \wedge H)}{P(S\wedge H)} \:=\:\frac{7}{10}\)

. . \(\displaystyle P(G\wedge S \wedge H) \:=\:\frac{7}{10}P(S\wedge H) \:=\:\frac{7}{10}\cdot \frac{5}{12} \quad\Rightarrow\quad P(G\wedge S \wedge H) \:=\:\frac{7}{24}\)


Therefore: .\(\displaystyle P\left(\sim\!\big[G\wedge S \wedge H\big]\right) \;=\;1 - \dfrac{7}{24} \;=\;\dfrac{17}{24}\)