Student Solutions Manual for
Introduction to Probability
with Statistical Applications
Geza Schay
University of Massachusetts at Boston
,1.1.1.
✁ ✂ ✁✂ ✁✂ ✂ ✁
a) The sample points are and the elementary events are
✄ ☎ ✁ ✄ ✂ ☎ ✁ ✄✂ ☎ ✁ ✄✂ ✂ ☎ ✆
✄ ✂ ✁✂ ✁✂ ✂ ☎
b) The event that corresponds to the statement ✝ at least one tail is obtained✞ is .
✄ ✁ ✂ ✁✂ ☎ ✆
c) The event that corresponds to ✝ at most one tail is obtained✞ is
1.1.3.
a) Four different sample spaces to describe three tosses of a coin are:
✟ ✠ ✡ ✄ ✁ ✂ ✁ ✂ ✁ ✂ ✂ ✁✂ ✁✂ ✂ ✁✂ ✂ ✁✂ ✂ ✂ ☎ ✁
✟ ☛ ✡ ✄☞ ✁ ✌ ✁ ✍ ✁✎ ☎ ✁
✟ ✏ ✡ ✄ ✑ ✑ ☎ ✁
an even # of s, an odd # of s
✟ ✒ ✡ ✄ ✁ ✂ ✁ ✂ ✁ ✂ ✂ ✁ ✂ ✁ ✂ ✂ ✁ ✂ ✂ ✁ ✂ ✂ ✂ ✁
✂ ✁✂ ✂ ✁✂ ✂ ✁✂ ✂ ✂ ✁✂ ✂ ✁✂ ✂ ✂ ✁✂ ✂ ✂ ✁✂ ✂ ✂ ✂ ☎ ✁
where the fourth let-
ter is to be ignored in each sample point.
✟ ✠
b) For the event corresponding to the statement ✝ at most one tail is obtained in three
✄ ✁ ✂ ✁ ✂ ✁✂ ☎ ✟ ☛ ✄✍ ✁ ✎ ☎ ✁ ✟ ✏
tosses✞ is . For it is and in it is not possible
✟ ✒
to ✓ nd such an event. For the event corresponding to the statement ✝ at most one tail is
obtained in the ✓ rst three tosses✞ is
✄ ✁ ✂ ✁ ✂ ✁ ✂ ✂ ✁ ✂ ✁ ✂ ✂ ✁✂ ✁✂ ✂ ☎ ✆
c) It is not possible to ✓ nd an event corresponding to the statement ✝ at most one tail is obtained
in three tosses✞ in every conceivable sample space for the tossing of three coins, because some
sample spaces are too coarse, that is, the sample points that contain this outcome also contain
✟ ✏ ✑
opposite outcomes. For instance, in above, the sample point ✝ an even # of s✞ contains
✂ ✁ ✂ ✁✂ ✂ ✂ ✂ ✁
the outcomes for which our statement is true and the outcome
for which it is not true.
1.1.5.
In the 52-element sample space for the drawing of a card
✡ ✡
a) the events corresponding to the statements ✔ ✝ An Ace or a red King is drawn,✞ and ✕
✡ ✄✗ ✟ ✁✗ ✁✗ ✘ ✁✗ ✙ ✁✚ ✁✚ ✘ ☎
✝ The card drawn is neither red, nor odd, nor a face card✞ are ✖
✡ ✄✍ ✙ ✁✜ ✢ ✣ ✤ ✢ ✣ ✥ ✢ ✣ ✦✧ ✢ ✣ ★ ✩ ✣✜ ✩ ✣ ✤ ✩ ✣ ✥ ✩ ✣ ✦✧ ✩ ✪
and ✛ , and
b) statements corresponding to the events
✫ ✬ ✭✮ ✯ ✣✰ ✯ ✣✱ ✯ ✣ ✲ ✯ ✪ ✬ ✭★ ✢ ✣ ✜ ✢ ✣ ✤ ✢ ✣ ✥ ✢ ✣ ✦✧ ✢ ✣ ★ ✩ ✣ ✜ ✩ ✣ ✤ ✩ ✣ ✥ ✩ ✣ ✦✧ ✩ ✪ ✬
, and ✳ are ✴
✵ ✶ ✬ ✵
The Ace of hearts or a heart face card is drawn, and ✷ An even numbered black card is
✶
drawn.
1.1.7.
Three possible sample spaces are:
✩ ✸ ✬ ✭ ✪ ✣
The 365 days of the year
1
,✩ ✬ ✭ ✪ ✣
January, February,. . . , December
✩ ✁ ✬ ✭ ✪ ✂
Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday
1.2.1.
✭ ✦ ✣ ✄ ✣ ☎ ✆ ✝ ✆ ✞ ✟ ✆ ✆ ✆ ✆ ✒ ✆ ✓ ✟ ✔
a) or ✠ ✡ ☛ ✡ ☞ ✌ ✍ ✎ ✏ ✍ ☞ ✑ ✏ ✌
✆ ✆ ✒ ✆ ✓ ✆ ☎ ✟
b) ✠ ✌ ✍ ☛ ✍ ☞ ✏ ✌ ,
✆ ✗ ✆ ✘ ✆ ✆ ✗ ✆ ✘ ✟
c) ✠ ✕ ✖ ✖ ✖ ✕ ✙ ✙ ✙ ,
✒ ✆ ✆ ✆ ✆ ✆ ✒ ✟
d) ✠ ✚ ✚ ✌ ✚ ✏ ✏ ✌ ,
✟
e) ✠ ✛ ☛ ✚ ✏ ✜ ✛ ✜ ✏ .
1.2.3.
✢ ✆ ✟ ✆ ✟ ✆ ✟ ✆ ✟ ✆ ✟ ✆ ✟ ✆ ✟ ✔
✠ ✣ ✠ ✤ ✠ ✥ ✠ ✣ ✤ ✠ ✣ ✥ ✠ ✤ ✥ ✠ ✣ ✤ ✥
1.2.5.
✦ ✧ ★ ✧ ✟ ✆ ✩ ✦ ✧ ★ ✪ ✧ ✆ ✓ ✟ ✧ ✆ ✆ ✒ ✆ ✝ ✟ ✟ ✆
✙ ☞ ✠ ✏ ✙ ☞ ✠ ✏ ✠ ✏ ✌ ☞ ✠ ✏
✦ ✧ ✩ ★ ✧ ✪ ✆ ✒ ✆ ✓ ✆ ☎ ✟ ✧ ✆ ✟ ✟ ✆ ★ ✧ ✩ ✦ ✧ ✪ ✆ ✆ ✓ ✆ ✫ ✟ ✧ ✆ ✒ ✟ ✟ ✔
✙ ☞ ✠ ✏ ✠ ✏ ✌ ☞ ✠ ✏ ✙ ☞ ✠ ✏ ✌ ✠ ✏ ☞ ✠ ✏
1.2.7.
✦ ✧ ✩ ★ ✬ ✪ ✆ ✒ ✆ ✓ ✆ ☎ ✟ ✧ ✆ ✆ ✒ ✆ ✓ ✆ ✫ ✆ ✝ ✟ ✆ ✒ ✆ ✓ ✟ ✆
a) ✙ ☞ ✠ ✏ ✠ ✏ ✌ ☞ ✠ ✏
✩ ✦ ✧ ★ ✪ ✬ ✆ ✓ ✟ ✬ ✆ ✆ ✒ ✆ ✝ ✟ ✆ ✆ ✒ ✆ ✓ ✆ ✝ ✟ ✔
but ✙ ☞ ✠ ✏ ✠ ✏ ✌ ☞ ✠ ✏ ✌
✦ ✧ ✩ ★ ✬ ✪ ✆ ✒ ✆ ✓ ✆ ☎ ✟ ✧ ✆ ✆ ✒ ✆ ✓ ✆ ✫ ✆ ✝ ✟ ✆ ✒ ✆ ✓ ✟ ✆
b) ✙ ☞ ✠ ✏ ✠ ✏ ✌ ☞ ✠ ✏
✩ ✦ ✧ ★ ✪ ✬ ✩ ✦ ✧ ✪ ✆ ✓ ✟ ✬ ✆ ✒ ✟ ✆ ✒ ✆ ✓ ✟ ✔
and ✙ ☞ ✠ ✏ ✠ ✏ ☞ ✠ ✏
✩ ✦ ✧ ★ ✪ ✬ ✆ ✓ ✟ ✬ ✆ ✆ ✒ ✆ ✝ ✟ ✆ ✆ ✒ ✆ ✓ ✆ ✝ ✟
c) ✙ ☞ ✠ ✏ ✠ ✏ ✌ ☞ ✠ ✏ ✌
✩ ✦ ✬ ✪ ✧ ✩ ★ ✬ ✪ ✆ ✆ ✒ ✆ ✓ ✆ ☎ ✆ ✝ ✟ ✧ ✆ ✆ ✒ ✆ ✓ ✆ ✫ ✆ ✝ ✟ ✆ ✆ ✒ ✆ ✓ ✆ ✝ ✟ ✔
and ✙ ✙ ☞ ✠ ✏ ✌ ✠ ✏ ✌ ☞ ✠ ✏ ✌
1.2.9.
✦ ✧ ★ ✢ ✔
The Venn diagram below illustrates the relation ☞ Using the region numbers from
✦ ✧ ★ ✆ ✒ ✟ ✧ ✆ ✒ ✟ ✒ ✟ ✆ ✦
the diagram, we have ☞ ✠ ✌ ✠ ✏ ☞ ✠ which is the region outside both
★ ✔ ✦ ✬ ★ ✆ ✒ ✟ ✬ ✆ ✒ ✟ ✆ ✆ ✒ ✟ ✆
and Similarly, ☞ ✠ ✌ ✠ ✏ ☞ ✠ ✏ ✌ ☞ the whole sample space.
✖
✦ ✧ ★ ✢
Figure 1. ☞
2
, 1.2.11.
✦ ★ ✆ ✦ ✆ ★ ✔ ✦ ✬ ★
1. Assume that that is, that whenever ✛ ✁ then ✛ ✁ Then ☞
✦ ★ ✟ ★ ★ ✟ ★ ✔ ★ ✦ ✬ ★ ✔
✠ ✛ ☛ ✛ ✁ or ✛ ✁ ✠ ✛ ☛ ✛ ✁ or ✛ ✁ ☞ On the other hand, clearly
✦ ★ ✦ ✬ ★ ★ ✔
Thus, implies ☞
✦ ✬ ★ ★ ✆ ✦ ★ ✟ ★ ✔
2. Conversely, assume that ☞ that is, that ✠ ✛ ☛ ✛ ✁ or ✛ ✁ ☞ Hence, if
✦ ✆ ★ ✆ ✦ ★ ✔
✛ ✁ then ✛ must also belong to which means that
✂ ✦ ✦ ✬ ★ ✆ ✦ ✬ ★ ★ ✆
Alternatively, by the de nition of unions, and so, if ☞ then substituting
★ ✦ ✬ ★ ✦ ✬ ★ ★ ✦ ★ ✔
for in the previous relation, we obtain that ☞ implies
1.3.1.
a) The event ✄ corresponding to ☎ ☞ ✆ ✤ is 4 or 5✝ is the shaded region consisting of the fourth
✂ ✂ ✩ ✆ ✞ ✪ ✓ ✆ ☎ ✞ ✆ ✆ ✔✔ ✔✆ ✫ ✟
and fth columns in the gure below, that is, ✄ ☞ ✠ ✤ ☛ ✤ ☞ and ☞ ✏ ✌ .
Figure 2. ✄ corresponding to ☎ ☞ ✆ ✤ is 4 or 5✝
✗ ✬ ✆ ✂
b) The event corresponding to ✟ or ☎ is ✄ the shaded region in the gure below, that is,
✩ ✆ ✞ ✪ ✩ ✆ ✆ ✔ ✔✔ ✆ ✫ ✆ ✞ ✆ ✆ ✒ ✪ ✩ ✓ ✆ ☎ ✞ ✓ ✆ ☎ ✆ ✫ ✪ ✟
✠ ✤ ☛ ✤ ☞ ✏ ✌ and ☞ ✏ ✌ or ✤ ☞ and ☞ .
3
Introduction to Probability
with Statistical Applications
Geza Schay
University of Massachusetts at Boston
,1.1.1.
✁ ✂ ✁✂ ✁✂ ✂ ✁
a) The sample points are and the elementary events are
✄ ☎ ✁ ✄ ✂ ☎ ✁ ✄✂ ☎ ✁ ✄✂ ✂ ☎ ✆
✄ ✂ ✁✂ ✁✂ ✂ ☎
b) The event that corresponds to the statement ✝ at least one tail is obtained✞ is .
✄ ✁ ✂ ✁✂ ☎ ✆
c) The event that corresponds to ✝ at most one tail is obtained✞ is
1.1.3.
a) Four different sample spaces to describe three tosses of a coin are:
✟ ✠ ✡ ✄ ✁ ✂ ✁ ✂ ✁ ✂ ✂ ✁✂ ✁✂ ✂ ✁✂ ✂ ✁✂ ✂ ✂ ☎ ✁
✟ ☛ ✡ ✄☞ ✁ ✌ ✁ ✍ ✁✎ ☎ ✁
✟ ✏ ✡ ✄ ✑ ✑ ☎ ✁
an even # of s, an odd # of s
✟ ✒ ✡ ✄ ✁ ✂ ✁ ✂ ✁ ✂ ✂ ✁ ✂ ✁ ✂ ✂ ✁ ✂ ✂ ✁ ✂ ✂ ✂ ✁
✂ ✁✂ ✂ ✁✂ ✂ ✁✂ ✂ ✂ ✁✂ ✂ ✁✂ ✂ ✂ ✁✂ ✂ ✂ ✁✂ ✂ ✂ ✂ ☎ ✁
where the fourth let-
ter is to be ignored in each sample point.
✟ ✠
b) For the event corresponding to the statement ✝ at most one tail is obtained in three
✄ ✁ ✂ ✁ ✂ ✁✂ ☎ ✟ ☛ ✄✍ ✁ ✎ ☎ ✁ ✟ ✏
tosses✞ is . For it is and in it is not possible
✟ ✒
to ✓ nd such an event. For the event corresponding to the statement ✝ at most one tail is
obtained in the ✓ rst three tosses✞ is
✄ ✁ ✂ ✁ ✂ ✁ ✂ ✂ ✁ ✂ ✁ ✂ ✂ ✁✂ ✁✂ ✂ ☎ ✆
c) It is not possible to ✓ nd an event corresponding to the statement ✝ at most one tail is obtained
in three tosses✞ in every conceivable sample space for the tossing of three coins, because some
sample spaces are too coarse, that is, the sample points that contain this outcome also contain
✟ ✏ ✑
opposite outcomes. For instance, in above, the sample point ✝ an even # of s✞ contains
✂ ✁ ✂ ✁✂ ✂ ✂ ✂ ✁
the outcomes for which our statement is true and the outcome
for which it is not true.
1.1.5.
In the 52-element sample space for the drawing of a card
✡ ✡
a) the events corresponding to the statements ✔ ✝ An Ace or a red King is drawn,✞ and ✕
✡ ✄✗ ✟ ✁✗ ✁✗ ✘ ✁✗ ✙ ✁✚ ✁✚ ✘ ☎
✝ The card drawn is neither red, nor odd, nor a face card✞ are ✖
✡ ✄✍ ✙ ✁✜ ✢ ✣ ✤ ✢ ✣ ✥ ✢ ✣ ✦✧ ✢ ✣ ★ ✩ ✣✜ ✩ ✣ ✤ ✩ ✣ ✥ ✩ ✣ ✦✧ ✩ ✪
and ✛ , and
b) statements corresponding to the events
✫ ✬ ✭✮ ✯ ✣✰ ✯ ✣✱ ✯ ✣ ✲ ✯ ✪ ✬ ✭★ ✢ ✣ ✜ ✢ ✣ ✤ ✢ ✣ ✥ ✢ ✣ ✦✧ ✢ ✣ ★ ✩ ✣ ✜ ✩ ✣ ✤ ✩ ✣ ✥ ✩ ✣ ✦✧ ✩ ✪ ✬
, and ✳ are ✴
✵ ✶ ✬ ✵
The Ace of hearts or a heart face card is drawn, and ✷ An even numbered black card is
✶
drawn.
1.1.7.
Three possible sample spaces are:
✩ ✸ ✬ ✭ ✪ ✣
The 365 days of the year
1
,✩ ✬ ✭ ✪ ✣
January, February,. . . , December
✩ ✁ ✬ ✭ ✪ ✂
Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday
1.2.1.
✭ ✦ ✣ ✄ ✣ ☎ ✆ ✝ ✆ ✞ ✟ ✆ ✆ ✆ ✆ ✒ ✆ ✓ ✟ ✔
a) or ✠ ✡ ☛ ✡ ☞ ✌ ✍ ✎ ✏ ✍ ☞ ✑ ✏ ✌
✆ ✆ ✒ ✆ ✓ ✆ ☎ ✟
b) ✠ ✌ ✍ ☛ ✍ ☞ ✏ ✌ ,
✆ ✗ ✆ ✘ ✆ ✆ ✗ ✆ ✘ ✟
c) ✠ ✕ ✖ ✖ ✖ ✕ ✙ ✙ ✙ ,
✒ ✆ ✆ ✆ ✆ ✆ ✒ ✟
d) ✠ ✚ ✚ ✌ ✚ ✏ ✏ ✌ ,
✟
e) ✠ ✛ ☛ ✚ ✏ ✜ ✛ ✜ ✏ .
1.2.3.
✢ ✆ ✟ ✆ ✟ ✆ ✟ ✆ ✟ ✆ ✟ ✆ ✟ ✆ ✟ ✔
✠ ✣ ✠ ✤ ✠ ✥ ✠ ✣ ✤ ✠ ✣ ✥ ✠ ✤ ✥ ✠ ✣ ✤ ✥
1.2.5.
✦ ✧ ★ ✧ ✟ ✆ ✩ ✦ ✧ ★ ✪ ✧ ✆ ✓ ✟ ✧ ✆ ✆ ✒ ✆ ✝ ✟ ✟ ✆
✙ ☞ ✠ ✏ ✙ ☞ ✠ ✏ ✠ ✏ ✌ ☞ ✠ ✏
✦ ✧ ✩ ★ ✧ ✪ ✆ ✒ ✆ ✓ ✆ ☎ ✟ ✧ ✆ ✟ ✟ ✆ ★ ✧ ✩ ✦ ✧ ✪ ✆ ✆ ✓ ✆ ✫ ✟ ✧ ✆ ✒ ✟ ✟ ✔
✙ ☞ ✠ ✏ ✠ ✏ ✌ ☞ ✠ ✏ ✙ ☞ ✠ ✏ ✌ ✠ ✏ ☞ ✠ ✏
1.2.7.
✦ ✧ ✩ ★ ✬ ✪ ✆ ✒ ✆ ✓ ✆ ☎ ✟ ✧ ✆ ✆ ✒ ✆ ✓ ✆ ✫ ✆ ✝ ✟ ✆ ✒ ✆ ✓ ✟ ✆
a) ✙ ☞ ✠ ✏ ✠ ✏ ✌ ☞ ✠ ✏
✩ ✦ ✧ ★ ✪ ✬ ✆ ✓ ✟ ✬ ✆ ✆ ✒ ✆ ✝ ✟ ✆ ✆ ✒ ✆ ✓ ✆ ✝ ✟ ✔
but ✙ ☞ ✠ ✏ ✠ ✏ ✌ ☞ ✠ ✏ ✌
✦ ✧ ✩ ★ ✬ ✪ ✆ ✒ ✆ ✓ ✆ ☎ ✟ ✧ ✆ ✆ ✒ ✆ ✓ ✆ ✫ ✆ ✝ ✟ ✆ ✒ ✆ ✓ ✟ ✆
b) ✙ ☞ ✠ ✏ ✠ ✏ ✌ ☞ ✠ ✏
✩ ✦ ✧ ★ ✪ ✬ ✩ ✦ ✧ ✪ ✆ ✓ ✟ ✬ ✆ ✒ ✟ ✆ ✒ ✆ ✓ ✟ ✔
and ✙ ☞ ✠ ✏ ✠ ✏ ☞ ✠ ✏
✩ ✦ ✧ ★ ✪ ✬ ✆ ✓ ✟ ✬ ✆ ✆ ✒ ✆ ✝ ✟ ✆ ✆ ✒ ✆ ✓ ✆ ✝ ✟
c) ✙ ☞ ✠ ✏ ✠ ✏ ✌ ☞ ✠ ✏ ✌
✩ ✦ ✬ ✪ ✧ ✩ ★ ✬ ✪ ✆ ✆ ✒ ✆ ✓ ✆ ☎ ✆ ✝ ✟ ✧ ✆ ✆ ✒ ✆ ✓ ✆ ✫ ✆ ✝ ✟ ✆ ✆ ✒ ✆ ✓ ✆ ✝ ✟ ✔
and ✙ ✙ ☞ ✠ ✏ ✌ ✠ ✏ ✌ ☞ ✠ ✏ ✌
1.2.9.
✦ ✧ ★ ✢ ✔
The Venn diagram below illustrates the relation ☞ Using the region numbers from
✦ ✧ ★ ✆ ✒ ✟ ✧ ✆ ✒ ✟ ✒ ✟ ✆ ✦
the diagram, we have ☞ ✠ ✌ ✠ ✏ ☞ ✠ which is the region outside both
★ ✔ ✦ ✬ ★ ✆ ✒ ✟ ✬ ✆ ✒ ✟ ✆ ✆ ✒ ✟ ✆
and Similarly, ☞ ✠ ✌ ✠ ✏ ☞ ✠ ✏ ✌ ☞ the whole sample space.
✖
✦ ✧ ★ ✢
Figure 1. ☞
2
, 1.2.11.
✦ ★ ✆ ✦ ✆ ★ ✔ ✦ ✬ ★
1. Assume that that is, that whenever ✛ ✁ then ✛ ✁ Then ☞
✦ ★ ✟ ★ ★ ✟ ★ ✔ ★ ✦ ✬ ★ ✔
✠ ✛ ☛ ✛ ✁ or ✛ ✁ ✠ ✛ ☛ ✛ ✁ or ✛ ✁ ☞ On the other hand, clearly
✦ ★ ✦ ✬ ★ ★ ✔
Thus, implies ☞
✦ ✬ ★ ★ ✆ ✦ ★ ✟ ★ ✔
2. Conversely, assume that ☞ that is, that ✠ ✛ ☛ ✛ ✁ or ✛ ✁ ☞ Hence, if
✦ ✆ ★ ✆ ✦ ★ ✔
✛ ✁ then ✛ must also belong to which means that
✂ ✦ ✦ ✬ ★ ✆ ✦ ✬ ★ ★ ✆
Alternatively, by the de nition of unions, and so, if ☞ then substituting
★ ✦ ✬ ★ ✦ ✬ ★ ★ ✦ ★ ✔
for in the previous relation, we obtain that ☞ implies
1.3.1.
a) The event ✄ corresponding to ☎ ☞ ✆ ✤ is 4 or 5✝ is the shaded region consisting of the fourth
✂ ✂ ✩ ✆ ✞ ✪ ✓ ✆ ☎ ✞ ✆ ✆ ✔✔ ✔✆ ✫ ✟
and fth columns in the gure below, that is, ✄ ☞ ✠ ✤ ☛ ✤ ☞ and ☞ ✏ ✌ .
Figure 2. ✄ corresponding to ☎ ☞ ✆ ✤ is 4 or 5✝
✗ ✬ ✆ ✂
b) The event corresponding to ✟ or ☎ is ✄ the shaded region in the gure below, that is,
✩ ✆ ✞ ✪ ✩ ✆ ✆ ✔ ✔✔ ✆ ✫ ✆ ✞ ✆ ✆ ✒ ✪ ✩ ✓ ✆ ☎ ✞ ✓ ✆ ☎ ✆ ✫ ✪ ✟
✠ ✤ ☛ ✤ ☞ ✏ ✌ and ☞ ✏ ✌ or ✤ ☞ and ☞ .
3
