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 3 times and lost 7 times.

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

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 324 times and lost 676 times.

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