Explanation
1. Analyze the Components
-
Total balls in bag: 6(Red)+4(Green)=10 balls.
-
Dice Outcome (i): A fair die has outcomes i∈{1,2,3,4,5,6}.
-
Probability of each die face: P(D=i)=1/6 for all i.
2. Condition for "All Red Balls"
Let E be the event that all selected balls are red.
The number of balls selected (i) depends on the die. Since there are only 6 red balls available, it is possible to select up to 6 red balls.
The probability of selecting i red balls given that the die shows i is:
3. Apply the Law of Total Probability
The total probability P(E) is the sum of probabilities for each possible die roll:
P(E)=i=1∑6P(D=i)×P(E∣D=i)
P(E)=61[(110)(16)+(210)(26)+(310)(36)+(410)(46)+(510)(56)+(610)(66)]
4. Calculate each term
-
i=1:106
-
i=2:4515=31
-
i=3:12020=61
-
i=4:21015=141
-
i=5:2526=421
-
i=6:2101
5. Summing the probabilities
P(E)=61[106+31+61+141+421+2101]
To add these, find a common denominator (210):
P(E)=61[210126+70+35+15+5+1]
Conclusion
The calculated probability is 1/5 (or 0.2). Since 1/5 is not listed in options (a), (b), or (c):
Correct Option: (d) None
