Suppose that in a senior college class of 500 students itis found that 210 smoke, 258 drink alcoholic beverages, 216 eatbetween meals, 122 smoke and drink alcoholic beverages, 83 eatbetween meals and drink alcoholic beverages, 97 smoke and eatbetween meals, and 52 engage in all three of these bad healthpractices. If a member of this senior class is selected at random,find the probability that the studenta.) smokes but does not drink alcoholic beverages;b.) eats between meals and drinks alcoholic beverages but doesnot smoke;c.) neither smokes nor eats between meals.Please show all steps to solve, not in books listed

Answer:a)Smoke but doesn't drink alcoholic beverages P=0.176  or 17.6%b)eats between meals and drinks alcoholic beverages but doesn't smoke P=0.062  or 6.2%c.) neither smokes nor eats between meals P=0.342  or 34.2%Step-by-step explanation:Using a Venn diagram with three circles, one for each bad health habit. Let us [tex]a[/tex] for the students that only smoke, [tex]b[/tex] for only drink alcoholic beverages, and [tex]c[/tex] for only eat between meals. Following this logic, [tex]d[/tex] represents smoke and drink alcoholic beverages but not eat between meals,  [tex]e[/tex] drink alcoholic beverages and eat between meals but not smoke,[tex]f[/tex] eat between meals and smoke but not drink alcoholic beverages. Finally [tex]g[/tex] for having all the three, this means [tex]g=52[/tex]. To find [tex]f[/tex], subtract 52 (the three problems) from 97 because this last number represents all the students that smoke and eat between meals, including the students that have the three bad habits. The same goes for 'd' and 'e'. [tex]f=97-52=45[/tex][tex]e=83-52=31[/tex][tex]d=122-52=70[/tex]To find [tex]a[/tex], subtract [tex]e[/tex], [tex]f[/tex], and [tex]g[/tex] from the total of smokers. This is because [tex]e[/tex], [tex]f[/tex], and [tex]g[/tex] represent smoke and at least another bad habit and [tex]a[/tex] represents only smoking. [tex]a=210-d-f-g=210-70-45-52=43[/tex]The same goes for [tex]b[/tex] and [tex]c[/tex]. [tex]b=258-d-e-g=258-70-31-52=105[/tex] [tex]c=216-e-f.-g=216-31-45-52=88[/tex]Adding all letters and subtract from the total to see if there is any healthy student: [tex]500-a+b+c+d+e+f+g =500-43+105+88+70+31+45+52=500- 434=66[/tex]a)Smoke but doesn't drink alcoholic beverages This will be 'a' (only smokes) and 'f'  ( smokes and eats between meals but doesn't drink) divided by the total of students. [tex]P=(43+45)/500=0.176[/tex]b)eats between meals and drinks alcoholic beverages but doesn't smoke This probability is 'e' divided by the total of students. [tex]P=31/500=0.062[/tex]c.) neither smokes nor eats between meals This will be 'b' (only drinks) plus the healthy students (66) divided by the total of students. [tex]P=(105+66)/500=0.342[/tex]