Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 4 times and lost 6 times.

1. Car
2. Goat
3. Goat
4. Car
5. Goat
6. Goat
7. Car
8. Car
9. Goat
10. Goat

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 348 times and lost 652 times.

1. Car
2. Goat
3. Car
4. Car
5. Goat
6. Car
7. Car
8. Goat
9. Goat
10. Goat
11. Goat
12. Goat
13. Car
14. Car
15. Car
16. Goat
17. Car
18. Goat
19. Car
20. Goat
21. Car
22. Car
23. Goat
24. Goat
25. Goat
26. Car
27. Car
28. Goat
29. Goat
30. Car
31. Car
32. Car
33. Goat
34. Goat
35. Car
36. Car
37. Car
38. Goat
39. Car
40. Car
41. Car
42. Car
43. Goat
44. Goat
45. Goat
46. Goat
47. Goat
48. Car
49. Goat
50. Car
51. Goat
52. Goat
53. Goat
54. Car
55. Goat
56. Goat
57. Goat
58. Goat
59. Goat
60. Goat
61. Goat
62. Car
63. Car
64. Goat
65. Car
66. Car
67. Goat
68. Car
69. Goat
70. Car
71. Goat
72. Car
73. Goat
74. Goat
75. Goat
76. Goat
77. Goat
78. Goat
79. Goat
80. Goat
81. Car
82. Car
83. Car
84. Goat
85. Goat
86. Goat
87. Goat
88. Car
89. Car
90. Goat
91. Car
92. Car
93. Car
94. Car
95. Goat
96. Goat
97. Goat
98. Goat
99. Goat
100. Car
101. Car
102. Car
103. Car
104. Goat
105. Goat
106. Goat
107. Goat
108. Goat
109. Goat
110. Goat
111. Goat
112. Goat
113. Goat
114. Car
115. Goat
116. Car
117. Goat
118. Car
119. Car
120. Car
121. Goat
122. Goat
123. Car
124. Car
125. Goat
126. Goat
127. Goat
128. Goat
129. Goat
130. Goat
131. Goat
132. Car
133. Goat
134. Goat
135. Goat
136. Goat
137. Goat
138. Goat
139. Car
140. Goat
141. Goat
142. Goat
143. Goat
144. Goat
145. Car
146. Car
147. Goat
148. Goat
149. Goat
150. Goat
151. Goat
152. Goat
153. Goat
154. Goat
155. Goat
156. Car
157. Car
158. Car
159. Car
160. Goat
161. Car
162. Goat
163. Goat
164. Goat
165. Goat
166. Car
167. Goat
168. Goat
169. Goat
170. Car
171. Goat
172. Goat
173. Goat
174. Goat
175. Goat
176. Goat
177. Goat
178. Goat
179. Car
180. Car
181. Goat
182. Goat
183. Goat
184. Goat
185. Car
186. Goat
187. Goat
188. Car
189. Car
190. Goat
191. Goat
192. Goat
193. Car
194. Goat
195. Car
196. Goat
197. Car
198. Goat
199. Car
200. Goat
201. Car
202. Goat
203. Goat
204. Car
205. Goat
206. Car
207. Goat
208. Car
209. Car
210. Goat
211. Goat
212. Car
213. Goat
214. Car
215. Goat
216. Car
217. Goat
218. Goat
219. Goat
220. Goat
221. Car
222. Car
223. Car
224. Car
225. Goat
226. Car
227. Car
228. Goat
229. Goat
230. Car
231. Car
232. Car
233. Goat
234. Goat
235. Goat
236. Car
237. Car
238. Goat
239. Goat
240. Car
241. Goat
242. Goat
243. Car
244. Car
245. Car
246. Car
247. Goat
248. Car
249. Goat
250. Goat
251. Car
252. Goat
253. Goat
254. Car
255. Goat
256. Car
257. Goat
258. Goat
259. Goat
260. Car
261. Goat
262. Car
263. Goat
264. Goat
265. Car
266. Car
267. Car
268. Goat
269. Goat
270. Goat
271. Goat
272. Goat
273. Goat
274. Car
275. Goat
276. Car
277. Goat
278. Goat
279. Car
280. Goat
281. Goat
282. Car
283. Goat
284. Goat
285. Car
286. Car
287. Car
288. Goat
289. Goat
290. Goat
291. Car
292. Goat
293. Goat
294. Car
295. Goat
296. Goat
297. Car
298. Goat
299. Goat
300. Car
301. Goat
302. Goat
303. Car
304. Goat
305. Goat
306. Car
307. Goat
308. Goat
309. Goat
310. Goat
311. Goat
312. Car
313. Goat
314. Goat
315. Car
316. Goat
317. Car
318. Goat
319. Goat
320. Car
321. Goat
322. Car
323. Goat
324. Car
325. Car
326. Goat
327. Car
328. Goat
329. Goat
330. Goat
331. Goat
332. Goat
333. Goat
334. Goat
335. Goat
336. Car
337. Goat
338. Goat
339. Goat
340. Goat
341. Goat
342. Car
343. Car
344. Goat
345. Goat
346. Goat
347. Goat
348. Goat
349. Car
350. Car
351. Goat
352. Goat
353. Goat
354. Goat
355. Car
356. Car
357. Car
358. Car
359. Goat
360. Car
361. Goat
362. Goat
363. Car
364. Goat
365. Goat
366. Goat
367. Goat
368. Goat
369. Goat
370. Goat
371. Goat
372. Goat
373. Goat
374. Goat
375. Goat
376. Goat
377. Goat
378. Goat
379. Goat
380. Goat
381. Car
382. Goat
383. Goat
384. Car
385. Goat
386. Goat
387. Goat
388. Car
389. Goat
390. Goat
391. Goat
392. Car
393. Goat
394. Goat
395. Goat
396. Car
397. Goat
398. Goat
399. Goat
400. Car
401. Car
402. Goat
403. Goat
404. Goat
405. Goat
406. Goat
407. Car
408. Goat
409. Goat
410. Car
411. Goat
412. Goat
413. Goat
414. Car
415. Goat
416. Goat
417. Goat
418. Goat
419. Goat
420. Goat
421. Car
422. Goat
423. Goat
424. Goat
425. Goat
426. Car
427. Car
428. Goat
429. Goat
430. Goat
431. Car
432. Car
433. Goat
434. Goat
435. Goat
436. Goat
437. Goat
438. Goat
439. Car
440. Car
441. Goat
442. Goat
443. Car
444. Car
445. Goat
446. Goat
447. Goat
448. Goat
449. Car
450. Goat
451. Goat
452. Goat
453. Goat
454. Car
455. Goat
456. Car
457. Goat
458. Goat
459. Car
460. Goat
461. Goat
462. Goat
463. Car
464. Car
465. Goat
466. Goat
467. Goat
468. Goat
469. Goat
470. Goat
471. Car
472. Goat
473. Goat
474. Car
475. Goat
476. Goat
477. Goat
478. Goat
479. Goat
480. Car
481. Goat
482. Goat
483. Goat
484. Goat
485. Goat
486. Car
487. Car
488. Goat
489. Goat
490. Car
491. Goat
492. Goat
493. Car
494. Goat
495. Car
496. Goat
497. Goat
498. Goat
499. Goat
500. Car
501. Goat
502. Car
503. Car
504. Goat
505. Car
506. Goat
507. Goat
508. Car
509. Car
510. Car
511. Goat
512. Goat
513. Goat
514. Car
515. Goat
516. Goat
517. Car
518. Goat
519. Goat
520. Goat
521. Car
522. Car
523. Goat
524. Goat
525. Goat
526. Goat
527. Car
528. Goat
529. Goat
530. Goat
531. Car
532. Car
533. Goat
534. Goat
535. Car
536. Goat
537. Car
538. Goat
539. Goat
540. Car
541. Car
542. Goat
543. Goat
544. Car
545. Goat
546. Goat
547. Car
548. Goat
549. Goat
550. Goat
551. Car
552. Goat
553. Goat
554. Goat
555. Car
556. Goat
557. Goat
558. Goat
559. Car
560. Goat
561. Car
562. Goat
563. Goat
564. Car
565. Goat
566. Goat
567. Goat
568. Goat
569. Car
570. Goat
571. Car
572. Goat
573. Car
574. Goat
575. Car
576. Car
577. Car
578. Goat
579. Goat
580. Car
581. Goat
582. Goat
583. Goat
584. Goat
585. Goat
586. Car
587. Car
588. Car
589. Goat
590. Goat
591. Goat
592. Car
593. Goat
594. Car
595. Goat
596. Goat
597. Car
598. Car
599. Goat
600. Goat
601. Car
602. Goat
603. Goat
604. Car
605. Car
606. Car
607. Goat
608. Goat
609. Goat
610. Goat
611. Goat
612. Car
613. Goat
614. Goat
615. Goat
616. Car
617. Goat
618. Car
619. Goat
620. Car
621. Goat
622. Car
623. Goat
624. Goat
625. Goat
626. Goat
627. Goat
628. Car
629. Goat
630. Goat
631. Car
632. Car
633. Goat
634. Car
635. Goat
636. Goat
637. Goat
638. Goat
639. Goat
640. Goat
641. Car
642. Goat
643. Goat
644. Goat
645. Goat
646. Car
647. Goat
648. Car
649. Goat
650. Goat
651. Goat
652. Car
653. Car
654. Goat
655. Goat
656. Goat
657. Car
658. Goat
659. Car
660. Goat
661. Goat
662. Goat
663. Goat
664. Goat
665. Goat
666. Goat
667. Goat
668. Car
669. Goat
670. Goat
671. Goat
672. Car
673. Goat
674. Car
675. Goat
676. Car
677. Car
678. Goat
679. Car
680. Car
681. Car
682. Goat
683. Goat
684. Goat
685. Car
686. Car
687. Goat
688. Goat
689. Goat
690. Goat
691. Goat
692. Car
693. Car
694. Goat
695. Car
696. Car
697. Goat
698. Car
699. Car
700. Car
701. Car
702. Car
703. Car
704. Goat
705. Goat
706. Goat
707. Car
708. Goat
709. Goat
710. Goat
711. Goat
712. Goat
713. Goat
714. Car
715. Goat
716. Goat
717. Goat
718. Goat
719. Goat
720. Car
721. Goat
722. Goat
723. Goat
724. Car
725. Goat
726. Car
727. Goat
728. Goat
729. Car
730. Car
731. Goat
732. Goat
733. Car
734. Car
735. Goat
736. Car
737. Goat
738. Car
739. Goat
740. Goat
741. Goat
742. Goat
743. Car
744. Goat
745. Car
746. Goat
747. Goat
748. Goat
749. Goat
750. Goat
751. Car
752. Goat
753. Goat
754. Car
755. Car
756. Goat
757. Goat
758. Goat
759. Goat
760. Goat
761. Goat
762. Goat
763. Goat
764. Goat
765. Car
766. Goat
767. Car
768. Goat
769. Car
770. Goat
771. Goat
772. Car
773. Goat
774. Car
775. Car
776. Goat
777. Car
778. Goat
779. Car
780. Goat
781. Goat
782. Goat
783. Goat
784. Goat
785. Goat
786. Goat
787. Goat
788. Car
789. Goat
790. Car
791. Goat
792. Goat
793. Goat
794. Car
795. Goat
796. Goat
797. Car
798. Car
799. Car
800. Goat
801. Goat
802. Car
803. Goat
804. Goat
805. Goat
806. Car
807. Goat
808. Goat
809. Goat
810. Goat
811. Goat
812. Goat
813. Goat
814. Goat
815. Goat
816. Car
817. Car
818. Car
819. Car
820. Goat
821. Car
822. Car
823. Car
824. Goat
825. Goat
826. Goat
827. Goat
828. Goat
829. Goat
830. Car
831. Car
832. Car
833. Car
834. Goat
835. Goat
836. Goat
837. Car
838. Goat
839. Goat
840. Goat
841. Car
842. Goat
843. Goat
844. Car
845. Goat
846. Car
847. Goat
848. Goat
849. Car
850. Goat
851. Goat
852. Car
853. Car
854. Goat
855. Goat
856. Goat
857. Goat
858. Goat
859. Goat
860. Car
861. Car
862. Goat
863. Goat
864. Goat
865. Goat
866. Goat
867. Goat
868. Goat
869. Goat
870. Goat
871. Goat
872. Car
873. Goat
874. Goat
875. Car
876. Goat
877. Goat
878. Goat
879. Car
880. Goat
881. Goat
882. Car
883. Car
884. Goat
885. Goat
886. Goat
887. Goat
888. Car
889. Goat
890. Car
891. Goat
892. Goat
893. Goat
894. Car
895. Goat
896. Car
897. Car
898. Goat
899. Goat
900. Goat
901. Goat
902. Goat
903. Car
904. Car
905. Car
906. Car
907. Goat
908. Goat
909. Goat
910. Goat
911. Goat
912. Car
913. Goat
914. Goat
915. Goat
916. Goat
917. Car
918. Goat
919. Goat
920. Goat
921. Goat
922. Goat
923. Goat
924. Goat
925. Goat
926. Car
927. Goat
928. Goat
929. Goat
930. Goat
931. Goat
932. Goat
933. Goat
934. Goat
935. Car
936. Goat
937. Goat
938. Goat
939. Car
940. Goat
941. Car
942. Goat
943. Goat
944. Car
945. Car
946. Goat
947. Goat
948. Goat
949. Goat
950. Goat
951. Goat
952. Car
953. Car
954. Goat
955. Goat
956. Car
957. Goat
958. Goat
959. Car
960. Goat
961. Goat
962. Goat
963. Goat
964. Car
965. Car
966. Car
967. Goat
968. Car
969. Car
970. Goat
971. Goat
972. Car
973. Goat
974. Car
975. Goat
976. Car
977. Car
978. Goat
979. Car
980. Car
981. Goat
982. Goat
983. Goat
984. Goat
985. Car
986. Goat
987. Goat
988. Car
989. Goat
990. Goat
991. Car
992. Goat
993. Car
994. Car
995. Goat
996. Car
997. Car
998. Car
999. Goat
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 110 times and lost 890 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 34 times and lost 66 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}