A box of 8 gloves contains three left-handed gloves and five right-handed gloves

Question

A box of 8 gloves contains three left-handed gloves and five right-handed gloves. Complete parts (a) through (d).

a. If two gloves are randomly selected from the box without replacement, what is the probability that both gloves selected will be right-handed?

b. If two gloves are randomly selected from the box without replacement, what is the probability there will be one right-handed glove and one left-handed glove selected?

c. If three gloves are selected with replacement, what is the probability that all three will be left-handed?

d. If you were sampling with replacement, what would be the answers to (a) and (b)?

Explanation / Answer

Answer to part a)

There are total 8 gloves: 3 left handed , 5 right handed

.

P(first glove is right handed) = 5/8

Now it is not replaced back , so total right handed gloves left is 4 , and total number of gloves is 7

P(Second glove is right handed) = 4/7

.

P(both right handed) = P(first right handed) * P(second right handed)

P(both right handed) = 5/8 * 4/7 = 5/14 = 0.3571

.

Answer to part b)

There are two cases to it : eight first is right handed and second is left handed , or , first is left handed and second is right handed

.

P(Case I) = P(first right handed) * P(second lefthanded)

P(case I) = 5/8 * 3 /7 = 15/56

.

P(Case II) = P(first left handed) * P(second right handed)

P(case II) = 3/8 * 5 /7 = 15/56

.

P(one right handed and one left handed) = P(case I) + P(case II)

[We add the two cases above , because either case I occurs or case II occurs ]

.

P(one right handed and one left handed) = 15/56 + 15/56

P(one right handed and one left handed)  = 30/56 =0.5357

.

Answer to part c)

P(three left handed gloves) = 3/8 *3/8 *3/8

.

in this case everytime the probability of left handed glove is same because when we select it ones, we replace it back , by this the total number of gloves remains same

P(three left handed gloves with replacement) = 27 /512 = 0.0527

.

Answer to part d)

Answer to a, with replacement would be : P(2 right handed gloves) = (5/8) * (5/8) = 25/64

Answer to b , with replacement would be: 2 * 5/8 * 3 /8 = 30/64

[2 is mutiplied in this part because there are two cases based on the order of right and left handed gloves]

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